Try Caldonazzo. It's in the north of Italy and you can access the Dolomites from there. They have one or two cheap hotels, and if you want to be even cheaper, book your hostel room in Trento and stay in Trento. From Trento, you take a local train north to Caldonazzo.
Or, you could stay in Trento and stay at the hostel (like $12 a night) or stay at Hotel Venezia for about $60 per night (for 2 folks) and just ride from Trento around the northern parts of Trento (ie: if you go from Trento to Lago di Garda, it's a steep hill directly out of Trento on the local roadway that can be a bit challenging, but fun if you're not dragging a ton of gear with you!).
You could also take the train into Trento, then switch to a local train and go to the town of Cles. I can dig out the name of this guy that owns an apartment, and I think he said he'd charge like $20 per person per night for use of his apartment, and from there, you can get right into riding in the Dolomites and do some great local hill riding and experience some great rides.
You could also take the train from Milan to Verona (or ride Milan to Verona if you can find route 15), then stay in Verona at the hostel. I like Verona, but the hostel, although nice, sucks because they kick you out between 9am- 5pm, so on days when you just wanted to loaf around and skip the ride, you'd still be homeless. But Verona has a few albergos that are nice and under 70 euro a night (for two folks) you can stay at, and they will have a place for you to store your bikes that is secure also. The hostel is secure too- they have iron gates around the perimeter that no one can sneak over. You could ride around Verona, which is hilly outside the city center, and you can certainly access the Dolomites from there (roadway 15 to Lago di Garda, then jump into the Dolomites and mountains from Lago di Garda).
You can also fly into Milan and either bike to the Como region or take a train to the Como region. But once in Como, you can stay at several of the hostels located throughout Lake Como (cheaper ones are further up by ferry from Como itself). Then you can ride around the Como region and into Switzerland. I heard the ride into Lugano is very challenging. I did the ride from Milan to Bellagio- very hilly and challenging, although the amount of enclosures along the highway may freak you out. So make sure you have lots of bright lights so people can see you in the tunnels! And once you get to Bellagio, you can ride right into Lugano- it's about 35 km.
There's definitely a lot of ways you can do it, but it starts with flying into Milano (MXP) and taking a train from there.