I am travelling to Italy this summer with 3 friends for 2 weeks.
They key is if you are going to rent a car or not.
1. Always get to hotels and ask for half pension. It means you get breakfast, dinner and room. This is the best way to save money.
2. Bormio has basically 2 or 3 rides.
Bormio - Stelvio -Switzerland Loop - Stelvio - Bormio
Bormio - Mortirollo (=paso foppa) - Gavia - Bormio.
Hotel Danielle in Bormio is well located not expensive.
3. Kastelruth This is a great town in sud tyrol. Very close you will find basically most of the famous cols of the Dolomites, Pordoi, Sella and of course the classic of them all, Tre Cima di Lavoredo. You can have a great loop starting in Cortina going south and turning North to Tre Cima. You can rehearse the famous cycling movie with Eddy in the Giro of 1974.
Another day you can do Passo Fedai (Marmolada) another classic with Falzarego.etc . You can spend here 5 days.
4. An interesting area of Italy with also famous climbs and ot too many tourist is North of Turin close to the border with France. You can do here Col d'Agnelo cross to France and do Col l'Izoard and back. etc. You stay in Sampiere.
5. My advice is rent a car, this will make you trip very efficient and you will be able to climb many of the famous passes.
It is hard to help you in general terms because the options are many. I assumed you want to climb the famous passes.