Here is my take on the original question...
1) Cyanotype is the typical 'gateway' into alternative process printing. The chemicals are inexpensive. The solutions needed are simple to make and the processing trivial. Van Dyke Brown prints are almost as simple and therefore also a good choice. However, being a silver based process it is a bit more expensive than cyanotype which is iron-based.
There are multiple variations of cyanotype but the classic version is probably a good place to start, especially if you use dilute acid (vinegar or citric acid) for the first wash instead of plain water.
If you want to get started without having to weigh solids and mix solutions, kits for these process (and many others) are available. Bostick & Sullivan and the Photographers Formulary are the two main suppliers.
2) Initially, use a paper that is made for alternative process printing or one that is known to work well with the process you choose without needed to pre-treat it. Using a variety of papers is part of the fun of alt process printing. However, this aspect is best left to explore once you can make prints reliably.
The alt process specific papers (Hahn. Platinum Rag, Legion Revere Platinum and Arches Platine, are the 'big ones') are relatively expensive but will eliminate one variable (i.e. source of failure) from your learning process.
A good source of information on paper is Chris Anderson's 'massive paper chart', see:
https://www.alternativephotography.com/massive-paper-chart/. The paid version is well worth the small price.
3) Buy or make a printing frame with a split back, it is especially useful for printing out process such as cyanotype and Van Dyke. There is lots of good information regarding printing frames, here:
https://www.alternativephotography.com/diy-contact-printing-frames-for-alt-photo/ and here:
https://www.alternativephotography.com/buy-printing-frame-reviews/
4) Initially, use the sun as a light source for exposure. You can't beat the price. If you want an artificial source, consider 495 nm LEDs. Again, lots of good info at:
https://www.alternativephotography.com/diy-uv-light-boxes-alternative-photography/
5) Have you noticed a pattern in then last few items?!
https://www.alternativephotography.com/ is a source of much great information about alt process printing.
Other good sources are books by James (
https://www.amazon.com/Book-Alternative-Photographic-Processes/dp/1285089316) which is, alas priced like a textbook or one by Enfield (
https://www.amazon.com/Enfields-Guide-Photographic-Alternative-Processes/dp/1138229075). There are others but those are the ones I am most familiar with.
Also very good are the series of books (see:
https://www.routledge.com/Contempor...rnative-Process-Photography/book-series/CPAPP) edited by Chris Anderson. These books are narrower in scope than those cited above but are very, very good.
6) As for a working space, one does not need a darkroom nor does one need running water.
Remember that many of these processes were invented in the last half of the 19th century before indoor was common. I have made cyanotypes at a remote cabin with no electricity nor running water. I hauled water from the lake.
One does not even need a particularly dim room these days.
What one needs is a space that excludes UV light (sunlight and fluorescent lights), a source of water and a place to get rid of waste water. Some counter or table space is also very handy.
I generally work in my basement with all of the lights ON! The lights are warm LED bulbs which replaced compact fluorescent bulbs in the ceiling fixtures. I haul water/waste water to and from my basement in five gallon buckets.
I wash prints in non-running water. Usually six trays moving prints from tray to tray every 5 or 10 minutes depending on the process. Agitation is intermittent; meaning when I remember to do it between other tasks!
This space works for all of the processes I use: cyanotype (including toning), salted paper (my main process), platinum/palladium and cuprotype.
My final recommendation...
Buy a copy of Chris Anderson's cyanotype book and start there. If you get tired of the 'blue" buy the book in her series written by Annette Golaz on toning cyanotypes with botanical materials.
Experiment and have fun. Every practitioner of alt process printing adapts things to their own circumstances and space. There are no definitive practices in this realm. One key to success is repeatability. Make careful notes as you work so you can reproduce your successes and avoid repeating mistakes. As you work out the kinks, change only a single variable at a time. Did I mention... have fun!