I will replacing a 12-26T cassette with a 12-32T cassette. Will I need a new chain as well? What adjustments might I need to make to the derailleur? Thanks.

Yes because your current chain is probably too short.
Regarding the derailleur, there is a screw that pushes against your hanger. You may have to adjust it so your cage plate will accommodate the larger low gear cassette cog. Basically the top jocky wheel should be close, but not rubbing the cassette.

Going from 26 to 32 will probably require a different rear derailleur. If you provide info with what bike you have and the specs, you'll get more accurate info. Here's a basic doc: Rear derailleurs buying guide | Wiggle Guides. If you google "long cage vs short cage" you'll get lots of info. Are you going to install the new cassette? If your LBS is going to do it, they would know if you need longer chain or derailleur.

Thanks for the advice. I was going to do it. I ordered the cassette and it occurred to me that it may involve more than just swapping out one cassette for the other. The new cassette is the same make and model as the old one - just a different range - so I figured the derailleur should work on both as the cable length between gears should be the same. I guess when the cassette comes in, I'll swap it out and see if it works or not and go from there.

If you have a long cage it's OK. Shimano's site gives you the range each derailleur is compatible with. Easy to check.