Renting an apartment that you plan to live in for awhile is a pretty big undertaking, and there are lots of obvious things you should look for (water damage, mold, etc.) When perusing an apartment, especially one that's not a newer-build model, do the following non-obvious things:
1) Flush the toilet. If it's slow/weird/gurgling/not refilling properly for some reason, take note.
2) Check the shower pressure. There are few things worse than signing a lease and realizing your shower is awful. Yes, the apartment staff is supposed to fix it, but in an older apartment there's usually nothing they can do about it.
3) Run the taps on full-blast hot and see what you get. Lukewarm water? You can probably adjust your own water heater once you move in. However, if the water runs cold after 5-10 minutes, the tank is probably tiny or malfunctioning -- not good.
4) Bathroom fan. This one's not as important to some people, but I like to make sure it exists and works properly.