Rabat means suburb in old Arabic and this is indeed because Rabat is the town lying outside the citadel of Mdina. The two towns used to be a single one until the Arabs dug a moat across the hill and fortified its tip creating Mdina. Rabat had walls of its own in Roman times but this can only be seen today in parts of the streetpattern.
There are several buslines going to Rabat, from the main busterminal in Valletta it should take about 35 mins. 53 Valletta-Rabat, 52 Valletta-Dingli (through Rabat), 51 Valletta-Mtarfa (through Rabat)
Rabat is also only a short walk from Mdina - through regular city streets with plenty of quaint buildings and homes. Simply exit the old capital's fortress walls and keep walking; soon enough, you'll be in Rabat.
St. Pauls' church with St. Paul's grotto, where the saint apparently lived after being shipwrecked on his way to being put on trial in Rome. Pope John Paul II visited this place of pilgrimage in 1990. One can take the underground passage from here to the Wignacourt Museum.
The Roman Museum which houses the remains of a Roman villa. It lies in between Rabat and Mdina. This rebuilt Roman villa features mosaics and sculpture from the original house excavated on the site, along with artifacts and displays relating the the Roman, Byzantine and Arab periods, from the 3rd century B.C. through the 10th century A.D.
The catacombs of St. Paul and St. Agatha, these are early Christian underground cemeteries, particularly the latter is very interesting to visit. Earlier descriptions refer to a "grumpy tourguide", but by June 2011 there is a friendly one who makes a special effort with children!
The Wignacourt museum, this was formerly Grand Master Wignacourt's palace, and is now a dusty but charming old-style museum. It now houses a mediocre local art collection on the ground floor and a slightly random collection of historic furniture and paintings upstairs. Do see the painting of Mary proferring her bared breast to her child with a strangely lascivicious look on her face. Also do visit the cellars of the museum which connect to another set of ancient catacombs, these have not been 'streamlined' for tourism like the other catacombs which make them particularly interesting.
Of course there are the usual souvenir shops, the Roman Villa Souvenir Shop (on the corner near the Roman Museum) has a vast selection at reasonable prices (such as €3.50 for well-made large metal souvenirs rather than the usual plastic tat).
Crystal Palace Bar,  St. Paul's Street, Rabat - A little hole in the wall just opposite the Roman Villa, you'd be forgiven for passing this little gem by completely. It is a typical Maltese bar, full of locals savouring the island's favourite savoury snack - 'pastizzi' (parcels of filo pastry filled with either ricotta cheese or a mushy pea mix). You can either sit down at one of the tables to enjoy pastizzi and tea (served in the obligatory pyrex tumbler!), or else order your food to go. Not one to miss!