For just starting out, I recommend sticking with one chemistry supplier -- that is, all Kodak, all Ilford, or whatever. The reason is that there are sometimes inconsistencies and gaps in instructions when you mix-and-match brands, leaving you scratching your head -- should you follow A's instructions or B's? Sticking with one supplier should help reduce confusion in an endeavor that's likely to be confusing enough when you just get started. Once you've got your sea legs, so to speak, you'll be better able to mix brands, since you'll have a better idea of what function everything performs, why things are timed the way they are, etc. With that knowledge you'll be able to fill in the gaps and resolve inconsistencies in instructions.
For specific products, I'd say to pick a popular general-purpose developer. Don't get too hung up on this choice, though; among mainstream developers, differences are pretty subtle, at least for a beginner. Getting something popular means you'll be better able to get help if something goes wrong. FWIW, Kodak's D-76 (which is virtually identical to Ilford's ID-11) is reputed to be the most popular film developer, but others that have been mentioned in this thread are also quite popular, so you'd be safe with any of them. Once you've picked a developer, buy the rest of the chemistry from the same outfit. For a print developer, Kodak's Dektol is quite popular, but Ilford and others also make popular products. Note that stop bath and fixer can be the same for both film and paper. (Hypo clear, too, if you use it with paper -- it's helpful for fiber-based papers but is generally not needed for resin-coated papers.)
One more comment: Agfa (maker of Rodinal film developer and the Neutol line of paper developers) has recently gone under. A company called (IIRC) A&O has bought up the rights to Agfa chemistry and will reportedly be manufacturing it. Agfa products are currently in short supply and so aren't a good choice at this time, simply because they'll be hard to find and even if you find some, you might not be able to find more if you use your first batch up quickly.