Letterblade

Things to check:
-If there is laundry in the building, look at it (don't just take the guy's word, and also then you'll know how dangerous or not the basement stairs are).
-Things to ask the current tenants, if they are around: what the utilities run, how long does it take the landlord to return a repair call.
-Things to ask the agent if they're not: Type and age of heating system (oil is much more expensive than gas, newer is more efficient, radiators might clunk/hiss but forced air is evil if you've got allergies, etc),
-Do all the windows have shade/blinds (these are required in Cambridge at least)
-Open all closets (good to know if there is a water line running through it before hand)
-Outdoor space and storage: shared or private

Things to keep in mind:
-Don't let them rush you
-The rules are different if it's a landlord or an agent showing the place
-If it's an agent, find out what the fee is, and it never hurts to ask if the landlord would split it
-Ask what the total due upfront is (find out if it's due at lease signing vs. move in)

I know it's daunting but you'll do fine.