The four-and-a-half necessary chemicals for B/W development are, in order of use:
- developer which makes the latent image visible;
- stop bath, which stops development promptly and protects the fix from the developer and which can be water;
- fixer, which gets rid of the undeveloped silver in the film and thus makes the image permanent;
- water to rinse the rest of the chemistry out of the film;
- optionally wetting agent aka photo-flo.
Of these (not in order).
- Stop bath is weak acetic acid and can be reused many times. The stop-bath you can buy has an indicator in it which is yellow if it's good or purple if it's exhausted: this is just something which is measuring the acidity of the solution. Make sure you dilute the stop bath you buy. You can use water for stop bath, but it's so cheap that it's worth buying the proper one. Stop bath essentially kills the developer which stops development quickly, and also means that dev does not leak into the fixer which preserves the fixer. Stop bath has no effect on the image.
- Fixer makes the image permanent, and can also be reused many times. Unlike stop bath there's no indicator although you can test fixer by fixing bits of undeveloped film and seeing if it clears fog. I don't do this but just fix for much longer than I need to and try to keep track of how many films have been through a given batch. There are different kinds of fixer (rapid, hardening &c &c), but I just use Hypam for everything. Fixer makes no difference to the image.
- Water: you really must wash films carefully after processing or the fix sitting in the film will eat it over time.
- Wetting agent / photo-flo: this is a detergent which you can add to the final rinse and which changes the surface tension to reduce drip marks on dried film. It may also contain an anti-fungal agent. This lasts for ever (you use basically one drop per film, I've only had one bottle ever) and is worth having.
And finally developer. There are hundreds of B/W developers, some of which are clones of each other (Kodak's D-76 and Ilford's ID-11 are pretty much the same, there are lots of Rodinal clones &c &c), some of which are use-once, some of which are not, and so on and so on. Many of them will have different contrast / grain / speed effects on the film. They can mostly be used at different dilutions and temperatures. There simply is no rule for what to use: the way you find out what to use is by experiment. Developer is the thing which actually matters in terms of what the photographs look like, and developer choice is a matter of taste.
There is a lovely web site, the massive dev chart, which lets you look up film / developer / time combinations. For your film, TMAX 400, it lists 306 possibilities with 82 different developers. There just is no single answer to this question.
To answer this question in general you need to do research and experimentation: there is a lot written about B/W developers, a lot of photographers have strong opinions and you need to look at people whose prints you like and find out what they used, and then experiment until you find out what works for you.
However, a good place to start would be to use Kodak's TMAX developer, which will almost certainly work well with TMAX 400. You can also look at the Kodak website where they have a number of documents about processing their films. They will have information in particular on the capacity of the dev and how long it can be kept once mixed.
The massive dev chart gives 16 options for TMAX 400 in TMAX dev, depending on dilution and desired film speed. Half of those are at ISO 400, and many of them will have come from Kodak's recommendations, with which you should compare them.
There is an app corresponding to the massive dev chart which contains the chart and also will work as a timer. You want this if you are processing B/W film. (Note I have no affiliation with the massive dev chart people, but you do want it.)