First time darkroom experience, what am I doing wrong?

This is getting ridiculous. You show a link to a safelight bulb of sorts and then talk about it going into the enlarger. Nope.
Oh, and it looks like there is no drawer above the negative carrier, unlike the Cadet II. You will need a below-the-lens kit and filters, could get expensive fast. You might be better off looking for a new, less basic enlarger.

That bulb isn't for the enlarger.
It is for the room - it is a safelight bulb.
You might expect it to be used in a larger darkroom - something like a school darkroom.
And then only in a lamp housing that either directs or attenuates the output.
If you put that in one of the clamp lights I referred to earlier and point it up toward a ceiling that is at least six feet above the lamp and the paper, the light will end up travelling at least twelve feet (to the ceiling and back) which would give you enough distance. If you paint that ceiling a medium gray or red, you can probably work with shorter distances.

BGriffin23 said:

a large lens, flat on one side and hemispherical on the other.

Click to expand...

That's a condensor. If you Google a bit on "condensor vs diffusion enlarger"; it'll start to make sense.

glbeas said:

It does sound like the safelight is the problem. Next box of paper try treating it like color paper and do it all in the dark. If this works get a better safelight and be sure to adhere to minimum distances the safelight needs to be from the paper.

Click to expand...

I agree. Do this sequence...

1. If you leave the safelight turned off, for pulling out a sheet and then process it and the result in absolute white base, that proves zero exposure of the paper and no light leakage from the room.
2. If you leave the safelight turned off, pull out one sheet and exposre it to the enlarger, the white border proves the continued shielding of paper from any light, even if the exposed area is too dense...then you know that the ONLY thing is adjusting exposure time and/or selected aperture of the enlarging lens.
3. If you turn the safelight turned on, pull out one sheet and exposre it to the enlarger, a resultant blackened border of the print proves the safelight is the culprit!

Here's a website where you can get a PDF of the manual. There is a filter holder. The filter hold tray sits ABOVE the condenser and under and away from the heat absorbing glass

http://www.jollinger.com/photo/cam-coll/manuals/manuals.html

I'm hoping this PDF file gets to you

The enlarger you have is apparently a condenser type. They're brighter for the same bulb power than diffusion enlargers (the diffuser absorbs some light). Assuming it's a common globe style bulb, however, 75W in the enlarger should be fine. A condenser enlarger will also generally produce higher contrast and sharpness from the same negative, compared to a diffuser type.

As others have noted, the safelight bulb you bought (apparently brighter than my similar one) needs to be distanced from your easel, paper safe or box, and trays to avoid fogging. I've always used mine in a reflector unit, bounced off the ceiling, which adds distance as well as absorbing some of the light. You can get a reflector clamp lamp at the local Big Box store for six to eight dollars, that will let you direct the light where it will light up the whole room without fogging everything.

75W enlarger bulbs are very bright. A PH140 in the 23XLII gives exposures 10-20 sec for me at f16 and 8x10 (Ilford MG classic w/C2 filter, 35 mm negative). l use it at f8 and add an ND4 filter, so that I don't shoot at minimum aperture. It varies by negative, but this is typical.

BGriffin23 Have a look at witw's #54 that is a very effective summary of what his tests will tell you which is the best method for getting to the truth about what's wrong.

Do this first which can be done right now and then and only then start a discussion about what you need to do with safelights' strengths, distances from enlarger etc
Until you do this you are in danger of going down paths that will not be direct routes to the cause

pentaxuser

My new bulb, suggested to my by Donald Qualls in reply # 20, and a new box of 25 sheets Ilford Multi-Grade 5x7 arrived today from Freestyle Photo. I had the afternoon off so was able to try them out. Starting off I tried the suggested tests in reply # 54 and came up with the following, developing some of the older –suspect– IMG paper, for the full recommended 90 seconds (given the way I count probably longer) in 14:1 Ilford Multi-Grade Paper Developer, f/16 (dimmest my ancient enlarger will go) and 11 inches measured between the enlarger's lens and the top surface of the easel:

1. No safelight, no exposure, straight to development in pitch blackness. Result: All white.
2. Exposed one frame from my first, and so far only, roll of film, for 1 second, then developed in pitch blackness, no safelight involved at any time. Result: A much better picture than any achieved in my first go-around, but over-exposed looking (i.e. much brighter/whiter than the picture I thought I took). Definite white border around the picture.
3. Same as above, but with the new safelight on when pulling the paper out of the box, positioning on the easel and during development. My timer turns the light off when the enlarger is on. Result: Same as test # 2.

Note: the safelight was in a reflector pointed up toward the corner of ceiling farthest away from the enlarger, trays and paper storage area. The new SL is considerably dimmer than the older one. I did not use the old SL at any time.

From these tests I have tentatively concluded the first safelight is safe and all my old paper is not ruined. A few dollars for a second bulb is no big deal and having a little extra paper on hand can't hurt. I continued my darkroom time "safe" in the knowledge just mentioned for 14 more sheets and came to some new conclusions, for which the forum is welcome to comment on:

1. Most images look fairly good with 2 seconds enlarger exposure at f/16 and 11". 1 second would produce a decent, but blown-out image and 4 seconds was recognizable but heading toward solid blackness.
2. Some film frames needed more time. One I tried for 1, 2, 4, 6 and 8 seconds (where 1 second = one push of the button on my timer which was set to 1 second) and I recon could have gone for 10 seconds and looked even better! Another film frame looked best at 3 seconds. Yet another could probably have had better results had I done another sheet for 3 seconds. Could this difference be due to the exposure levels of the film while in the camera? The 3 second exposure was from a frame that was noticeably dark and probably originally underexposed.
3. IQ difference between the Ilford Multi-Grade paper and the no-name Multi-Tone paper is hardly, if at all, perceptible, but the Multi-Tone is half the cost.

Finally I managed to get through to Beseler customer service and was informed my enlarger model is so old they cannot readily identify it and therefore no longer support it. The CS rep said my film carrier looked odd to her and she offered me the equivalent part for the Cadet II which given the measurements shown in the pictures she emailed me will likely not line the film frame up to the same center point as the carrier on my enlarger and so I have decided not to buy that replacement part for now.

Below I have attached some of the prints I developed today. Since I lack a scanner I had to use my digital camera to do the best I could. Forgive the terrible lighting. These pictures are:

778: The first test image, right after the blank white sheet, 1 second exposure.
779: 2 second exposure, Ilford Multi-Grade
780: 4 seconds
781: 6 seconds
782: 8 seconds
783: 8 seconds exposure, Multi-Tone
784: 2 seconds exposure, likely could have been improved at 3.
785: 2 seconds
786: 3 seconds

It sounds like you are making process.

BGriffin23 said:

Could this difference be due to the exposure levels of the film while in the camera? The 3 second exposure was from a frame that was noticeably dark and probably originally underexposed.

Click to expand...

Yes.
When you say a frame of film was dark, do you mean dense, or clear?
A dense negative will require more exposure at the printing stage.
A dense negative results from more exposure (not under-exposure) at the taking stage.
Speaking more generally, I prefer my printing times to be longer - at least 10 seconds. That gives time for dodging and burning. Some of the options that make a difference are lower power bulbs, neutral density in the light path, and always using a multigrade filter.
One final technical point: you will discover that a ten second exposure without interruption provides more exposure than ten one second exposures. That will become more important with experience, as repeatability becomes more important to you.

Dark = dense. Frame is mostly grey with few if any clear areas. So must have been overexposed. The whole logic of negatives takes some getting used to.

I will look into both the bulb and NDF ideas. This will be even more necessary for making even smaller prints (an idea which interests me) given that a closer enlarger = more intense light. I can already see opportunities for D&Bing in some of my prints.

I will be sticking to multiples of one- or two-second exposures for now for consistency.

BGriffin23 said:

Dark = dense. Frame is mostly grey with few if any clear areas. So must have been overexposed. The whole logic of negatives takes some getting used to.

I will look into both the bulb and NDF ideas. This will be even more necessary for making even smaller prints (an idea which interests me) given that a closer enlarger = more intense light. I can already see opportunities for D&Bing in some of my prints.

I will be sticking to multiples of one- or two-second exposures for now for consistency.

Click to expand...

Exposure is still too bright. Best starting exposure is around 8 seconds. It's fast enough for printing and gives enough options for dodging and burning.
From 8s it's easy to go one stop up or down and increase/shorten time if you need more speed or more control.

First thing is to establish workable exposure. Better strategy for exposure strip is to go with F stop scale.
More details here: https://unblinkingeye.com/Articles/TestExpo/testexpo.html

Under the lens multigrade filters would be the easiest for your enlarger: https://www.bhphotovideo.com/c/search?Ntt=multigrade filter&N=0&InitialSearch=yes&sts=ma
I had good luck finding filters in local ads. Ebay might be another source. Cheap dichroic (colour) enlarger is also an option if there is a cheap one. Every pack of Ilford paper used to come with filtering chart for colour enlargers.

Could I just hold one of these under the lens when exposing to get the right effect? Remember my enlarger is so ancient it has no provision for filters. So any "filtering" will have to be improvised.

https://www.bhphotovideo.com/c/product/114879-REG/Ilford_1762628_Multigrade_Filter_Set_3_5x3_5.html

Holding a multigrade filter under the lens works. There are also filter holders that fasten onto the lens with setscrews and hold the smaller filters, in frames, that are designed for those holders (or you can improvise your own). These are better, because they'll leave your hands free for dodging or burning, which you'll surely want after a bit.

The problem with holding the filters is that, unless they are mounted in a frame, they are likely to suffer from dirt and damage in relatively short order. They can't be easily cleaned, and dirt and damage will affect contrast and, in extreme cases, resolution.

BGriffin23 said:

Could I just hold one of these under the lens when exposing to get the right effect? Remember my enlarger is so ancient it has no provision for filters. So any "filtering" will have to be improvised.

https://www.bhphotovideo.com/c/product/114879-REG/Ilford_1762628_Multigrade_Filter_Set_3_5x3_5.html

Click to expand...

This is under the lens filter holder with filters. Many of these filters fade over time if they are too old.
https://www.bhphotovideo.com/c/product/24675-REG/Ilford_1762617_Multigrade_Filter_Set_with.html

One you linked to is much cheaper, but inconvenient to use without filter drawer or holder of some kind. I have few of both versions and never really used the one without under the lens holder and frames.

Other option is to 3d print or fabricate some kind of holder. Some thin plywood or rigid cardboard cut with fret saw or knife could work well.

Medium format slide mounts make good holders.

MattKing said:

Medium format slide mounts make good holders.

Click to expand...

Because, you know, everyone just has a box of these lying around...

Donald Qualls said:

Because, you know, everyone just has a box of these lying around...

Click to expand...

No - but they are available on the internet, and a single box is a lot cheaper than the difference between the small set of filters linked to, and the complete set of framed filters you can buy.
If the OP was local, I have a few awkward to use metal ones that I could give to him.