To double move you have to take the same move action twice in a row. -PHB 284
According to this rule it would appear that it is not legal to shift and then trade your standard for a walk or a run. You could shift and trade your standard for another shift. You could shift and use your standard action for a charge. You could also double run or double walk, but you would provoke an attack of opportunity.
However, further study reveals a bit of confusion:
On your turn you can move twice if you take a move action instead of a standard action. If you take the same move action twice in a row, you're taking a double move. -PHB 284
That "if" seems to imply that taking two move actions is not necessarily a double move. It would follow that a shift and then a run action IS allowed, but is not called a "double move" anymore.
The difference is kind of minute. A double move is set apart by the fact that the two actions are resolved as one action, allowing you to end the first action in an ally's square or the last square of your first move to enter difficult terrain because you are not technically stopping there.
If I interpreted this rule incorrectly, let me know.