If budget and price is your deciding factor then that will definitely guide some of your choices. I'll admit I'm not completely in touch with the hostel scene but Lucca is not the kind of town that will have a ton of cheap, youth-centric options. Pisa has universities and is more geared to that and as a big city Florence will have more options as well.
If you just want cheaper lodging you could look into someplace like Pistoia or Prato. Neither of them are very touristy so staying should have some cheaper options and they are midway between Lucca and Florence for visiting both on the cheap local train line if you're interested. Otherwise I would see if you can find a hostel cheap in Florence and visit these other places as day trips. Or a couple of days in Pisa - check out Lucca and maybe the coast of you want - and then onto Florence and check out Siena and maybe Arezzo from there by bus and/or train. Bologna is close and a huge university town so probably a lot of hints on cheap lodging, food etc there. The regionale train to Bologna is slower (1.5 hours versus 35 minutes or so) but is less than half the price.
Siena has a university and may have some cheaper options but it is heavily touristed so I don't know if most places will have been used into more expensive lodging versus cheap ones. The cheap options will probably be more of a room rental outside of town situation rather than in the city center.
Unfortunately touristy does not lend itself to having a bunch of cheap housing and dining options. Food in Italy is generally very cheap (and very good) but very touristy places like Florence are always looking to get as many tourist euros as they can. I've heard reports from Italians of having to argue for the "Italian price" in Florence when charged twice as much for things like pizza as they deemed reasonable. If you're traveling in a strict budget you may need to make some choices, but that doesn't mean you can't enjoy the country.
Enjoy Italy,
=Tod