When my dd was 15m we did a train journey from Innsbruck to Verona and she loved it. Italian trains are very comfortable and we upgraded to first class because the price wasn't much more.

Check out //www.seat61.com for information about Italian trains and how to book.

No, don't do it. First Sorrento is hugely overrated and the costa amalfitana is no fun when you are two unless you have a boat. Beaches are very small and crowded, and the water gets deep very quickly. At two you wouldn't appreciate the beauty and scenery of the landscape. Plus the road is nightmare if you are carsick.
Once in Perugia, go to the other coast, to Numana, beautiful beach, relaxing atmosphere and a true authentic Italian feeling.
June is the perfect time to be on the Adriatic sea. IF you want to stay near Rome, Sperlonga might be a good option , there is an express train to/from rome airport, and Sperlonga offers something for everyone, from history to lazy beach and great food.

BE aware that in Italy you pay to stay on the beach unless you of to a spiaggia libera (free beach), but it is not expensive, you get two sun beds, and a parasol, to which you can hang your bag, there are toilets, showers, and a cafe / restaurant which usually cooks great food. It will also be safer to leave the bag and go for a swim that leaving the bag on a towel. This said, leave valuables at the hotel and just go with a 50 euro note folded in your shorts/skirt pocket because better safe than sorry.

Train is fine but imho Sorrento is not very toddler friendly, hot cobbles, busy and no real beach. Might be better to head north.