You can use no filters for a rough G2 but the time will be approx half when using a filter. What this means is that if you decided you do need a filter to get to G1 or G3 then your unfiltered time is little use to you.. You will need to start the test strip process again with the filter. That's why people generally recommend when using 'Ilford' multigrade filters (or equivalent) use a filter at all times. If your enlarger has inbuilt filtration (Cyan, Magenta & Yellow) then you can either dial in the suggested amounts to maintain equal exposure, or dial in whatever you require and alter the time to suit.
As for the original problem... I suggest making sure you can get a full black on your paper 1st. Take a small piece out (don't forget to close the box) and turn the white lights on. Delevelop it for the full time. If it's not black then your paper is probably inkjet media, or your developer is not developer.
If that works, then test that you know which side of the paper is emulsion. What (manufacturer & surface) are you using? When did you buy it? Under safelights take another piece out and draw a line across it with a pen. Cut it in half and put one piece each way up (one with line showing, one without) under the enlarger. With no negative in and no filtration, blast them with lots of light... say 1min @ f4.5 on your lens. Develop them for the full time. One should be black, one white. Check the lines and see which one is on white (the other should be grey to black). That's the non emusion side. Examine them (gloss level, surface feel) and remember which is the emulsion side. As noted above, using gloss paper makes this step redundant as it's obvious. Some matt and even semi-matt papers are hard to see which is the emulsion side.
Once you've got this far, stick a G2 filter in the enlarger. Stop your lens down to f8. Insert a negative and focus. Since you know which side is the emulsion from the previous test, put your paper emulsion side up under the enlarger. Now make a test strip (covering the paper with some cardboard) with 5,5,10,20,40 secs steps. This gives you actual times on the paper of 5,10,20,40,80 seconds, or 5 full stops range. If one of them is not too dark then your enlarger probably has the wrong globe in it or extra neutral density in the light path. If you do have a range of grey sections, then work out which one is what. Write on each section with a pen 5,10,20,40,80. Decide which section is too light and which is too dark. Because the times are whole stops I'm pretty sure one will be too light and one too dark. Say the 2nd section is too light but the 3rd is too dark... that would put the exposure in the 10 to 20 secs range. Do another test strip with an initial exposure of the lower time (eg 10secs in this example) and strips that are the higher time - lower time divided by 5. So for this example, (20-10)/5=2secs. If it was one section more then (40-20)/5=4secs (so base exposure of 20secs and 4 sec strips). Once developed,you should have a good idea of a good time to do a full enlargement. Do a full enlargement next to see the whole image. This may need further tweaking as a base exposure + several timed bursts is not equal to one exposure and you may want to dodge and burn as well.
