I'm sure ScottW will jump in here, but definitely don't skimp on memory (say "RAM" to the salesperson). The more, the better. And ScottW will be more precise on the processor (say "CPU" to the salesperson), but don't let the salesperson talk you in to buying bells and whistles you don't need but will raise the price substantially. If you're going to buy a desktop, I would think a Celeron would do (I wouldn't say that for a laptop) because it's much cheaper, but I'll defer to ScottW on the exact kind you should get.
Most Machines these days come equipped with enormous hard disks (say "HDD" to the salesperson), so that shouldn't be an issue unless you want to store a lot of large file photos and videos on it.
As far as brands go, stick with the well known names, like Dell, HP, or IBM (now "Lenovo" I think, but I think they kept the IBM brand name). If you do get a clone knockoff, or even if you get a brand name but especially if you get a knockoff, you might ask the store about their warranty policy. Ask them if they'll send it off to a repair facility (they probably will unless they have skilled technicians right there . . . most don't, except maybe COMPUSA).
Ask them specifically what will happen if the thing doesn't work right out of the box (that doesn't happen very often, but it does happen). And say "right out of the box" . . . that will make you sound geeky.
And ask them how long it will take to get the machine back if it develops a flaw and you have to take it in. I'm sure they'll say something like "that will depend on the repair", but if it's a substantial repair they should just give you a new machine anyway (if it's still under warranty of course).
If you get a brand name, Dell, or HP, or IBM, or whatever you get, they may say that the original manufacturer will take care of service. If that's the case, then make sure and register the purchase with Dell, or HP, or IBM, or whatever you get.
If you have any friends who have made a computer purchase at the store, ask them how the store's service was before you make the purchase. And just one opinion is certainly not statistically significant or scientific, so the more people who have made computer purchases there that you can ask, the better.
And, even if you do your "due diligence" on the purchase (which you've already started to do by coming here and asking about it), bear in mind that you may get a lemon anyway. I once purchased an IBM computer and had nothing but trouble with it. Fortunately, IBM's service was good, their turnaround times were good, and I eventually ended up getting a new machine that was much more reliable than the first.
Write all the questions down you want to ask the store, and maybe memorize them. Because you don't want it to look like you're reading off a checklist that some geek friend of yours gave you. Speaking of geek friends, if you know of one that will go with you and that you trust, take them if they'll go. Often, another person will think of some question to ask that you haven't thought of. Take a friend with you anyway . . . a salesperson that plays games will be less likely to do so if there's another person with you. Plus, another person will be a "witness" and can attest to what the salesperson said in case you get into one of those "he said . . . she said" things.
Oh . . . one more thing. Check the store's reputation with the Better Business Bureau. You can do that online. The object there is to see how many complaints have been filed against the store with the BBB.
As I think of more things to do for "due diligence" I'll post back, but this is all I can think about off the top of my head right now.