You can drive from the center of Rome, along the Via Tiburtina. However, this is a slow, ugly road that passes through Rome's unattractive suburbs. A quicker route is to take the Autostrada A24 in the direction of L'Aquila. The exit for Tivoli is soon after the main toll station, where you have to take a ticket. However, you should avoid returning by the same route on a weekend evening, particularly during the ski season, as the toll station can get very congested.
Urban bus 4 goes from the city to the Villa Adriana. Don't be conned into thinking it's walkable. it's nearly four miles and road signs are not good. Besides you need all your time and energy at the Villa.
Where the Emperor Hadrian retreated for peace and quiet, Villa Adriana
Villa Adriana, Phone: (0039) 0774 530 203 (email: firstname.lastname@example.org),  Country retreat of the Emperor Hadrian and about equal in area to the old city of Pompeii. Located down the hill back towards Rome (you drive close to it on the Via Tiburtina). Accepts the Roma Pass. It is a very large estate (the largest Roman Villa known) with ruins in various levels of preservation. Entry is €10 for adults. The map given to you for the tour is not very useful. There are audio guides for €4 which require you deposit a ID with them (watch out they have been known to lose them). There are 20 or so pretty long audio snippets for some key places in the estate but each site also has a large sign which also has the same as the audio in text in four languages. Car parking is to the left as you approach the villa and at the entrance area there is a restaurant, a cafe and bookshop too.
Villa d'Este, (near the center of the town.), ☎ 199.766.166: from abroad: 0039.0424.600460 (email@example.com), . Closed on Mondays. Accepts the Roma Pass. This is a UNESCO World Heritage Site and a truly unique place. Although the villa is itself quite interesting, with some attractive frescoes on the ceilings, the main reason for visiting here is the terraced gardens below it. These were conceived and built between 1550 and 1572 by Cardinal d'Este. The numerous fountains, both large and small, rely on natural water pressure, which takes advantage of the steep slope on which the gardens are built. Water comes mainly from the River Aniene, which was partly diverted through the town. For a long time in the 18th and 19th Centuries the gardens were allowed to fall into a state of decay, but they were taken over by the Italian Government at the beginning of WW1 and have gradually been restored to something close to their original splendor. The gardens were said to have drawn their inspiration from Villa Adriana and, probably, quite a few of the statues used were stolen from there too. In turn, many were stolen from Villa d'Este when it fell into neglect. edit
Villa Gregoriana, Largo Sant'Angelo (walk/drive north from the train station), ☎ 06 39967701, . March & Oct 15-Nov M-Sa 10-1430 Su 10-16, April-Oct 15 M-Su 10-1830. A recently restored and reopened park on the River Aniene with the ruins of an ancient Roman temple and villa. There are several paths that meander through woodland with striking views of a series of waterfalls largely engineered by Pope Gregory XVI. A beautiful and relaxing place for a picnic with very few tourists. Last entrance 30min before closing time. Accepts the Roma Pass.€5/adults, €2.50/children, €4/in groups, €12/family. edit
Rocca Pia Castle, Vicolo Barchetto, ☎ 0774 313536. An impressive castle/keep in the centre of Tivoli. It is just east of Villa d'Este, west of the pedestrian bridge from the train station.edit