If you don't know much about the main things to see in Tivoli (Villa d'Este, Villa Adriana, etc) or the history of the town, I recommend buying a booklet with info on these things, either from a street vendor or a souvenir shop. That way, you can read it as you check out the sights.
What to buy: Vendors and shops also sell all sorts of other trinkets and souvenirs. Aside from the booklet, I also bought a painted fan for my collection.
On the way to Villa del Este you will find many shops and vendor street where you can find souvenirs, books and jewellery.
De camino a Villa del Este encontraras muchas tiendas y puestos en la calle donde se encuentra desde souvenirs, libros y joyeria.
In Piazzza Trento near the entrance to the Villa d'Esta are many stalls aimed at the tourists.
I like the look of all these ceramic plates....wish I'd bought one now ;-/