I'm going to list off a few things and they may not be coherent. I need to stay busy.
-location - make sure it is within reasonable distance from work or school or wherever you'll be going often
-does the apartment require renters insurance and in general how are the bills handled - are utilities consolidated or do you pay some sort of share
-maybe check the surrounding area out both during the day and at night and on various weekdays
-road ways next to apartment - traffic
-allow pets
-allow smoking
-some have pools, computer stations, exercise areas, etc - decide what you'd want/use
-which floor - top floor - have to climb stairs everyday and moving is harder. Bottom floor - people stomping/loud above you. Middle floor can have both those
-nearby food and stores - grocery store etc
-always go inside the actual apartment before getting it - sounds silly but...
-leasing options - is there a contract where you have to stay so long else you have to pay - can you sublease if needed
-fun th

-is it furnished or have to bring your own stuff
-fun things to do in the area
-Roomate may can split pay for nicer place, but then have roommate - ups and downs
-go with gut/instinct
I'm not sure any of that is helpful...more just a list of stuff. Eh, oh well

1) Always get something well within your budget, not a place you can barely afford - you never know when you'll have an expensive emergency and don't want to worry about making rent.
2) If possible, try to rent from an individual instead of a property company since a personal relationship can go a long way if you need to negotiate rent or repairs.
3) If you tour a place and have a bad/weird/off feeling, don't rent it. The right place will come along when you need it.
4) When touring, check for vermin under drawers, in cabinets, and behind fixtures. You might feel awkward being so thorough, but it'll save so much headache later.
5) Make sure you know your rights (google "[your country/state] tenant rights") and ensure that any lease/contract you sign doesn't violate those rights. If it does, that's a sign to find another place.
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I can answer more specific questions, those are just basic things that everyone should be aware of.

The set I'm currently looking at is in a part of town I'm not particularly fond of. It's on the opposite end of town than where I usually frequent, but that's only a 15-20 minute commute, so it's really not that bad.
-I'm not fond of the fact that it's right next to a Wal-Mart. :/ But the place is pet friendly, which is the main thing I'm looking for. I believe I'd have to supply my own furniture, but I'm okay with that. Major appliance are provided and there are washer/dryer hookups. I have access to a set if I need.
-I don't care so much about pools/gyms/etc.
-I believe it's non-smoking, and I don't smoke
-I can maybe afford a 2 bedroom on my own (they don't offer a single), but I can also get a roommate. It's a college town.
-most of the individuals to rent from around here won't allow pets, unfortunately...
-I'm not sure how utilities are handled at this place, so I'll have to check.

But yeah, main priority is pet friendly. Cat friendly, specifically. There are a bunch of places here that claim to be pet friendly, but really only allow dogs

Are you in P-Town?

Nah. Logan

If you already have a cat: many places that allow dogs may also allow cats, but just didn't bother to say so in their terms. After all, any damage or noise that a cat could do could also be done by a dog. I once talked a landlady into allowing a pet goose, so don't give up.
If you're planning to get a cat: really consider if you can handle a cat at this point: can you afford emergency vet visits? Are you going to be home often or busy at work/school? Are you planning to move again soon? I'm not trying to preach, just making sure you don't get into more than you can handle at a tumultuous time.

VERY IMPORTANT
When you find an apartment and movie in take photos of ANY damages you see before bringing in any boxes or furniture, show then to your landlord, and save the photos
That way when you move out, those damages wont be blamed on you and you have physical evidence of showing your landlord that you didnt make those damages