When I went to Athens, we used the Hop-On-Hop-Off buses. They're in almost every major city in Europe and are rather well-priced. I apologize that I don't remember all of the names, but I was there several months ago.
One of the stops is a history and statue museum, which I highly enjoyed and it is free for students.
If you go to the Acropolis, which has a lot of beautiful temples, I believe this was also free for students. We toured it on our own, but you can pay for a guided tour. One thing to note in that area is there are a lot of stray dogs, but most will just sun-bathe and pay no attention to you. Do still stay aware of them though, as two got into a fight as we were walking above the Theatre of Dionysus.
Just one stop after the Acropolis, if I recall correctly, is the Temple of Zeus. If you show your ticket to the Acropolis, then you get free entrance. Otherwise I believe it was 2 EUR. You won't spend as much time here, but it is certainly still worth it.
If you are looking for some souvenirs, then the Agora is right outside of the entrance of a main Metro station. It is a very large set up of both stores and tents with things for sale. This is also an excellent place for pictures, and the shop owners are very nice and are up for haggling for a price.
Another piece of advice is to find the best restaurants, look for the hole-in-the-walls. We wandered around shortly after arriving in the city and went into the first restaurant we saw and it was great!
We were only there for a few days, and one day was basically lost due to a very large strike, so this is by no means a complete list. I hope this answer helped!