You need a paper developer, a film developer, and a dual-purpose (film and paper) fixer.
In addition to these bare basics, stop bath, hypo clear, and a wetting agent help.
Here is an example of a basic Kodak list:
HC-110 film developer, 16 oz. syrupy concentrate. There are three "stages" of HC-110, each representing different dilutions: concentrate (undiluted "syrup" straight from the Kodak bottle), stock solution (intermediate solution), and working solution (the solution actually used to develop film). Use one oz. concentrate to 31 oz. water to make a quart of working-strength developer; you can skip the stock solution with a batch this large, as it is easy to measure 1 oz. of syrup (but harder to measure smaller amounts for smaller batches). For batches of working solution smaller than 1 qt., make up a stock solution first, which is 1:3 concentrate to water. Then, make working solution of any volume by using a 1:7 stock to water ratio.
(or)
D-76 film developer powder, pouch to make 1 gal. stock solution. Use straight and pour back after use, counting rolls processed and adding time every four rolls; or use 1:1 stock to water and dump after each use.
Dektol paper developer powder, pouch to make 1 gal. stock solution. Mix 1:2 stock to water to use. Keep used working solution after a printing session if it still works well, or discard it if it does not.
Flexicolor Fixer, 1 gal. Color fixer, but works perfectly for black and white and is ultra cheap. Non-hardening, which is preferable for most uses. Use 1:4 for film or 1:9 for paper. There is pretty much no reason to use anything else, unless you want a hardening fixer, in which case you can use the liquid Kodafix Solution, or powdered Kodak Fixer with the included hardener added.
Here is a basic Ilford list:
Ilfotec HC or ID-11 film developers. HC has the same mixing directions as HC-110, and ID-11 has the same ones as D-76. Developing times and developer characteristics are effectively identical as well. HC has become much more expensive than HC-110, so I no longer use it. I switched back to HC-110.
Ilford PQ or Ilford Multigrade paper developers. Same directions for both: mix 1:9 to use. These have the convenience of coming as a stock solution, as opposed to a powder like Dektol. PQ and MG are slightly different in contrast and hue. IME PQ is better for cleaner ("snappier") tonal separation in the whites and is warmer in hue (very slightly creamy), while MG is better for more soft and subdued high tones and is neutral in hue. PQ proves somewhat hard to find in stock in the U.S., and you may have to order a five liter container when you order it. PQ is my standard developer now. Once I tried it, and saw the slight differences compared to MG, I switched. It suits most of my pictures slightly better than MG.
Ilford Hypam or Ilford Rapid Fixer. Hypam works the same as Rapid Fixer. However, Hypam allows the addition of a hardener if desired, and is cheaper. However, Rapid Fixer is more commonly found in stock, for some reason. Even if using mostly Ilford chemicals, I would still consider the Kodak Flexicolor option for a fixer, as it is so cheap.
Both Kodak and Ilford make stops, wetting agents, and hypo clearing agents, however, in the U.S., Kodak's always seem to be in stock and cheap, and Ilford's seem hard to find and expensive. I suggest Kodak Indicator Stop Bath in the smallest bottle available ( 16 oz., I believe), Kodak Photo-Flo 200 in the smallest bottle available (again 16 oz., I believe), and Kodak Hypo Clearing Agent, which costs about four bucks for the quart pouch, which makes five quarts of working solution. It is much cheaper to buy the gallon pouches (to make five gallons of working solution), but I find that I never use the whole gallon of stock before its published shelf life has passed. I don't know how bad sodium sulfite in water can really go, but it is something to be aware of.