Maybe I can help out. (I moved into my first place in 2012, have since changed apartments too).
1.) Utilities. Are any utilities included? If so, which ones? Usually, water, trash, and sometimes heat are included. If you are lucky like me, you have an apartment which pays the electric too.
Ask about the oven/stove. Is it gas or electric? If it’s gas, ask if you have to pay for cooking gas. If it’s electric, it will either come out of your electric bill or, if your landlord pays the electric for you, then you don’t have to worry.
2.) How much are utilities every month? What is the average electric bill?
3.) What are the tenants like? Do you get a lot of students/families/working single professionals? I like to ask this because it gives you an idea of how much noise to expect. Families = loud (I try to avoid family-oriented apartments for this reason), Students may or may not be loud, depends if they attract partiers, and working single professionals are usually quiet because they are gone all day.
4.) Pets? If you are considering an animal, ask this before hand. Are pets allowed? Which ones? Is there an added security deposit or an added monthly fee because of a pet?
5.) Are appliances included? (ask about fridge and stove/oven). Most apartments will not include other appliances. Be prepared to bring your own microwave.
6.) Square footage? (If you haven’t seen it yet)
7.) Security features? Buzzer system? Locks?
8.) If you are looking to get yourself cable/internet or TV, ask if the apartment is cable ready.
9.) Laundry facilities? Is there laundry onsite? Where? What does it cost?
10.) Do you offer additional storage space (perhaps in the basement) for tenants?
11.) How do you pay rent? Is it via mailed check? Or do you offer an online payment system? Automatic withdrawal.
12.) Do you have income requirements/limits in order to live in this apartment?
13.) What is the parking situation? On street or off? Is there are a fee for off street parking?