Aaagh: you said 'resorts' .... well from Bergamo you will need to either hire a car or take a train to ... Lake Garda, the Italian Tyrol, Venice and via Milan for Florence, Pisa or Bologna.
From Rome train station, you can get to Naples, the Amalfi coast, Sorrento, Pompeii, Florence, Pisa, Venice ...
Have you investigated the Kilometrico ticket which allows for up to 6 of you to travel up to 2000 km for a set fee? Contact the Italian Tourist Board in London, or FS - Ferrovie dello Stato ... Italian State Railways.
For cheap hotels, there really aren't many, especially in the resorts above, but places to start include the local tourist offices (usually accessible on the web) or site such as Opodo, Expedia, Alpharooms or chains such as Ibis or Campanile. Good luck e buon viaggi!