You may have noticed that we did not opt for Hadrian's villa on this trip. It is under continuous repairs and when we visited many of the good parts were not open. If you have a notion to visit you should investigate this first. I'd check HERE. Note that the villa is a ways out of town. You'll end up needing a taxi I think, as the bus system appeared sluggish.
Another place I would have liked to visit was this.
Tempio della Tosse. It is a Roman structure quite near the Villa d'Este. Nobody really knows what it was originally, perhaps a tomb or a nymphaeum. In the Middle Ages it of course became a church. For reasons that are also obscure, it is a church associated with the relief of any disease that causes you to cough! Allergies to the plane trees in Italy this time of year would have made me an enthusiastic visitor. Alas, it is on private property. And the most recent owners do not appear to have done any more with it than any of the others since the day when our old pal Piranesi sketched it in the mid 1700s.
Tivoli in May did not have a "touristy" feel to it. There were some restaurants in swell locations - we ate at one that overlooked the temples - but it was mostly a local clientele. And there is a seldom visited "down town" Tivoli that had a lot of smaller dining establishments that looked appealing.
If you opt for the Parco Villa Gregoriana hike you will naturally want to do it in decent weather. We had a train to catch and dashed through in something like 90 minutes. It was too quick a pace. We did not have time for any of the side paths that lead to some spectacular waterfalls.
Villa d'Este is an hour or so in the gardens, spend a bit of time in the villa building also. It has been battered about and looted a few times, so there are very few furnishings, but the wall paintings are swell and have been nicely restored.
I almost never give lodging advice but our B and B host in Tivoli went above and beyond the call of duty. He figured we would have trouble finding the rather out of the way location and met us on the way there. Breakfast was great, up on the rooftop. The price was reasonable. If you plan to stay over night in Tivoli I therefore suggest: La Mensa Ponderaria
Tivoli turned out to be one of the favorite stops on our Italian trip. I recommend it either as a day trip or if you prefer, to stay over night. The train service is cheap and frequent. And with a bit of route planning you can skip Termini Station and make your trip via the Roma Tibertina station. It is new, clean, efficient and seems to be free of larcenous pick pockets.