A city of the Pheonician and Punic periods from the C6th BC it was the base of a powerful trading empire spanning the entire south Mediterranean and home to a population of the order of half a million people. Its most famous general was Hannibal who crossed the Alps to battle with the Romans. Hannibal suffered his first significant defeat at the Battle of Zama in 202 BC, which ended the 2nd Punic War. After over 50 years of being watched closely by Rome, they were eventually attacked in the 3rd Punic War. The citizens defended the city against the Republic of Rome in 146BC yet lost, and Punic Carthage was completely destroyed by the order of the Senate. The site was redeveloped by the Romans a century later and Carthage became the capital of the Roman province of Africa. A UNESCO World Heritage List site.
Take the TGM light rail line from Tunis to Carthage-Hannibal station.
Mostly it's best to walk. However, the area is large and sightseeing is sweaty work on a hot day.
If you are tired, you may wish to get a cab between some of the major tourist spots. This should be cheaper in low season.
For example, 10 TD (the driver offered 7, but he was nice and helpful and patient) bought me a trip in December 2010 from the museum down to the Amphitheatre, 10 minutes of the driver waiting around, a trip to the Water Cisterns (his suggestion, not calculated in the price), a further 5 minutes of the driver waiting around, and a trip down to Tophet.
Very little remains of Carthage today, and Punic Carthage in particular was completely obliterated. However, sections have been dug up.
You can buy a ticket for 9 DT (December 2010) that allows access to about 10 different sites, which are a big challenge to see in one day. Only some sites have extensive English signage - many, such as the museum, do not.
Map of the historic sites of Carthage
Antonin Baths. Ruins of the largest Roman baths outside Rome itself. The site also includes a Punic cemetary, some old houses, some Punic kilns, a chapel, some graves, mosaics, etc. Guides are available in a number of languages, and may be worthwhile as the site is large. TD 5 for entry (or use the multi-site ticket) + TD 1 for photo rights. Please note that it is illegal to take photographs in the direction of the presidential palace. Doing so, especially when traveling alone, may land you in jail for up to 3 years. (Note: Actually the guards were pretty relaxed when I visited, I don't think paranoia is necessary but obviously don't be over the top!) There is a small, reasonably priced cafe at the exit area closest to the water.
Carthage Museum, and the Acropolium (St. Louis Cathedral). Most remnants excavated from the ruins have been stored in the cavernous museum located on Byrsa Hill, documenting both the Punic and the Roman eras. 8:30AM-5:30PM (until 7PM in the summer), entry TD 4.2 or with the multi-site ticket. Note that as of December 2010 part of the museum is closed for the remaking of some exhibits. The museum grounds offer sweeping views of the coast and city, and also include the ruins of some Punic streets, the former site of a public library, numerous sculptures, a chapel or church, some excellent mosaics and some coffins. Unfortunately many items about the grounds are unlabelled, so a guide may be useful (or eavesdrop on a tour group if you can). Ignore the shifty guy trying to charge you 1 DT to use the toilet. St Louis Cathedral forms one edge of the museum, but did not appear to be open on my visit - it was completed in 1890.
World War II North Africa American Cemetery and Memorial,  Just north of Carthage, a five minute walk from the Amilcar Station. Open daily except for December 25 and January 1; 9AM-5PM. The Cemetery is the final resting place for 2,841 American military Dead lost during the War in North Africa. A monument is inscribed with the names of 3,724 Americans whose remains were never found or identified. The memorial court contains large maps in mosaic and ceramic depicting operations across Africa. Free.
Water Cisterns. A very large series of water cisterns which functioned as a water redistribution point at the terminus of a long (90km?) aquaduct from the hills to the south. The remnants of the end of the aquaduct are still standing. The site is free, no ticket is required. Also offers reasonable views of the city.free.
International Festival of Carthage in the summer is a treat with big name dancers, singers, and artists from all over the world for a very reasonable price (10.5 DT). Plastic chairs are not worth the price (26 DT) unless you are really aching for back-support. Most people bring cushions or blankets for the amphitheatre seats. Plan a late night out with concerts normally ending after 1AM.
There´s a lot of leather products to buy in Tunisia, from handbags to puffs. It´s cheaper than in Europe or America, and there´s a lot to choose from.
You can also buy many handmade items that are characteristic of Arab countries.
The Hotel Villa Didon, next to the main Carthage ruins, offers a stylish lounge and terrace with delightful views. Very pleasant for an evening beer or glass of wine.
Sidi Bou Said - this charming town is just a few train stops from Carthage and is a great place to eat and watch the sunset