Ciao Chilelily,
Where will you be starting your trip at? since you ask for help in the sequence, it would be useful to know where you're also starting...
If you're starting with 5T, maybe you're coming down from Milan/Genova... the best way to get there is by train. Monterosso is the first town starting from the north, then Vernazza, Corniglia, Manarola, ending with Riomaggiore. You can hike along the coast, but there are also trails further up in the hills to the very top along the crest of the mountains that separate the 5T from inland. They are all great trails, but as Elena pointed out, depending on when you're coming, some might be closed or off limits if weather has caused any slides along the way. This fall will be 2 years from the heavy floods that completely devasted Vernazza and Monterosso. While they have recovered mostly, it is the reason why fall can be a time when hiking might not be the best way to visit them... if that is the case, you'll have to stick to trains between the towns.
Then if you're heading south and inland, the order should be:
Lucca, Pisa, San Gimignano (take train to Empoli, then to Poggibonsi, from there take bus) and then on to Siena (with bus or train), leaving Florence last before heading to Bologna. You can complete most of travel by train except for San Gimignano.
Hope this info helps! If you tell us where you start, we might be able to provide other ideas if it doesn't work with what I've suggested so far ;-)
As far as budget accommodation, take a look
hostels but the best starting point is first the city, then the type of accommodation. Start here:
http://www.tuscanyaccommodation.com/
Don't leave out 1 - 3 stars, they are often very, very nice but at great prices as well as small, family run B&Bs.