By plane
Direct flights from Nairobi to Juba are offered on a variety of commercial airlines. It's not cheap - a US$500 round trip is the minimum you can expect to pay. Charter and UN flights are also available from Lokichokio. Nowadays, it is also becoming common to use a route through Addis Ababa. Ethiopian Airlines has two direct flights daily. Egyptair has one daily direct flight from Cairo.
There are several direct flights from Khartoum to Juba costing around US$200 each way; however, the airlines running this route keep changing AirWest is currently flying [Feb 2008], while Nova Airlines has shelved this route. There are also regular flights from Entebbe in Uganda twice a day.
Egypt Air flies direct from Cairo, hence providing a single airline route from London. Kenya Airways and Jetlink Express are also operating from Nairobi twice a day. Some other local airlines like Sudan Airways and Feeder Airlines also connect Juba to Malakal, Rumbek and other destinations in South Sudan.
By bus
If overlanding, daily buses run from Kampala. The ride takes 12 hours minimum if the roads are dry, but it only takes one broken down truck or deep flooded muddy pothole to add hours to the journey. Departures are at 3 a.m. Nile Coach near Gateway in the city center of Kampala with an overnight stop in Yumbe. No regular buses run to Kenya although some trucks run the route - most Kenyan traffic runs through Uganda as the roads are in better shape from there.
If flying down from Khartoum, bring your passport. Although you don't need a separate visa in addition to the standard Sudanese one, your passport will be checked for a Sudan visa at Juba airport.
If overlanding from Kenya or Uganda, it is still advisable to get your travel permit/visa for South Sudan GOSS office in Nairobi or Kampala, 100 USD and only a one-month, single-entry visa is currently available. This will save you a lot of hassle at the border.
Also: daily buses run from Kampala with Sudanese ownership LOL brand. About 12 hours. No overnight in Yumbe. Through Gulu and Atiak.
Unless cost is a BIG issue, take the plane from Entebbe to Juba. The bus is a great cultural journey, secure-but-dusty, and the southbound trip in daylight affords great views of the countryside. However, the northbound trip during night-time offers police checkpoints and pee stops in desolate places!