There is not much left of the glorious old days of Carthage. The Romans destroyed nearly everything they found here to build their own settlements. Thus most of what remains is of Roman origin. There are six main sights, spread out over a wide area.

The easiest way to visit all the sites is by taking the TGM (light rail) line that runs through the middle of the area to Carthage Hannibal Station. From there, walk uphill and turn left just over the top. This road will go uphill again and leads to the top of Byrsa Hill (it's about 500m walk from the Hannibal station).
Byrsa Hill gives good views of the whole area view from its peak. At its base is the Cathedral of St Louis, also visible from afar. It was built by the French in 1890 and dedicated to the 13th century saint-king who died on the shores of Carthage in 1270 during the 8th Crusade. It was deconsecrated and closed for years, but now it has been restored and is open to the public. The National Museum is the large white building at the back of the cathedral. This is where you can buy a ticket that gives access to all the sites around the area. Every time you visit another site, it is stamped on your ticket.

Walk downhill and in opposite direction of Carthage Hannibal Station to visit the Roman amphitheatre on the west side of the Byrsa. It is supposed to have been one of the biggest amphitheatres in the Roman Empire, but little is left of it nowadays. Most of its stones were used for later buildings. The huge cisterns northeast of the amphitheatre were the main water supply for Carthage during the Roman era. They are not worth a visit as they are completely ruined.

From Carthage Hannibal Station you go downhill and turn right to see the Antonine Baths. There is not much left, but just the size of the site is impressive. The Magon Quarter is near the water, a few blocks south of the baths. Recent excavations have revealed an interesting residential area.

A little uphill from the baths (crossing the TGM rail) you will find what is left of some Roman Villas.

The Sanctuary of Tophet was discovered in 1921. The Tophet was a sacrificial site with an associated burial ground, where the children of Carthaginian nobles were sacrificed to appease the deities Baal Hammon and Tanit. There is not much to see and it's hardly worth the effort.