Guadalajara has may things to offer, it is one of the most "Mexican" states. All the "charro" culture and tequila comes from this part of mexico. There is a train tour that will take you to Tequila, where all the distillers are and give you a tour and sampling.
Guanajuato would be my next stop. The city is host to the Cervantino international arts festival once a year and has many things to see like the "kissing" alley and the mummy museum. The city also has many tunnels dating back to the colonial period when it was a mining city.
Queretaro is also a very nice place to visit. It's a colonial city also and has a very nice downtown area where one can shop and relax
In Mexico City you can do anything and everything. It's one of the biggest cities in the world and it is very diverse in all aspects. You will not grow bored. Visit the Xocimilco canals. It's like a Mexican version of Venice.
In all these cities there are cheap hostels where you can stay. Just google "Mexican hostels" and you will find several. almost 5 years ago