All the major sites along the wall can be got to by public transport. In summer you can get the AD122 bus service to Hadrian's Wall, which travels between Newcastle and Carlisle along the line of Hadrian's wall. This service connects to Newcastle, Hexham and Carlisle.
There are very frequent train and bus services between Newcastle and Carlisle via Corbridge, Hexham and Haltwhistle. Hexham is a very pretty market town and is a mini transport hub for the centre of Hadrian's wall; Haltwhistle is only a few miles to the south of the wall while Corbridge is the site of a Roman town. For more local travel information check out the Bus and Metro page, this provides more details on how to get to Hadrian's Wall sites and includes useful links to local bus timetables for Hadrian's Wall.
If you are travelling to the North East of England, and are thinking of flying, then the most convenient airport is Newcastle airport, which has flights to many domestic and European destinations. The other airport that serves the North East of England is Durham Tees Valley airport, which is located just 24 miles South of Durham. If you are coming from further afield then Manchester Airport has a much wider range of international flights, otherwise you may need to travel to the via London Heathrow and then get a connecting flight to Newcastle (travelling via Amsterdam or Brussels are other possibilities). From London there is also a fast and frequent train service to Newcastle via York and Durham, while Carlisle is on the West Coast main line.
You can also travel from Europe to Newcastle by ferry. DFDS provides ferry services from Newcastle to Holland. This is great if you're bringing a car over as you will avoid the very long drive up from the South of England, and is also a good jump off point for the Lake district or a tour of Scotland.
Finally if your time is very short and you don't want to hire a car then several local taxi firms offer guided tours to the wall. This is especially useful if you have a small group. For example EcoCabs in Hexham can help you with trips from half a day to three days, Hadrian's Wall taxis also offer a similar service. Both companies have very friendly and helpful drivers.