Can't speak to Naples or Florence, as I've only passed through, but as for Rome there are plenty of quality hotels to choose from within your budget. If you wanted more privacy/space you may consider an AirBnB as you can often rent an entire large apartment for less than the cost of a hotel room. If you're sticking to trains, toddler in tow, you may want to steer clear of the Amalfi Coast as it requires multiple transfers from train to bus along the way - and they're often quite crowded. If you really want to see the coast (and it is magnificent), I suggest renting a car from Naples to streamline the process. From what I remember this will run you about 100 euro.
As far as food in Rome, Trattoria Settimio is a speakeasy type throwback to classic Rome. They keep the door locked and only let you in after the owner gives you a once-over and signals his approval. Expect the no frills basics - pasta, sauce, meatballs, wine. It isn't chic or contemporary, but its a unique experience, and it's the kind of place Don Corleone might eat if he was in town for a meal. If you're looking for something a little more modern, Verso Sera, outside Campo de Fiori, has unbelievable pastas and a great outdoor terrace for warm nights.
Edited by thecuff - 4/21/16 at 4:11pm