From Pakistan
The Karakoram Highway takes you from Kashgar to Gilgit via Tashkurgan, Hunza and Passu or vice versa. The Khunjerab pass is some 4800 meters high and is only open in summer. Border formalities are not very strict. There is one bus daily in eevry direction that takes two days to go from the border to Kasghar - you spend the night in Tashkurgan. In Pakistan it is pretty easy to find onward transport.

From Nepal
The border at Khasa between Nepal and Tibet is open. If you are coming from Nepal you need a permit to enter Tibet - it is not very likely they let you in if you just show up. The Nepalese issue visa upon arrival.

From Kyrgyzstan
From Kashgar in west China you can cross into Kyrgyzstan via the Torugart, it is a great trip, but you need to get a special permit. It is possible to arrange this permit in Kashgar but it takes a day or two and you may have to go in an officially organized trip which drops you off at the border. On the Kyrgyz side of the border you need to take a cab to the first town of some size, Naryn, which is still a couple of hours away.

Vietnam
The Chinese Vietnamese boredr is open. Ther are buses to Vietnamfrom Nanning.