In the US, if a defendant in a criminal trial (in state court) wants to change the venue of the trial, is it possible to have it changed to another state?
There is some room for change of venue from, say, one county to another, or one municipality to another, within the same state. In certain circumstances this may even be constitutionally required. See Sheppard v. Maxwell, 384 U.S. 333, 363 (U.S. 1966) ("where there is a reasonable likelihood that prejudicial news prior to trial will prevent a fair trial, the judge should continue the case until the threat abates, or transfer it to another county not so permeated with publicity"). But, there's no mechanism to move a state criminal trial to another state.
A cousin of the scenario you're talking about is where D has committed crimes in multiple states. In such a situation the prosecutors from different states (and also the federal system) may meet and co-ordinate their strategies, and part of that will be deciding who charges what and who goes first. An example of this is the DC sniper case, which I remember pretty vividly because I grew up in the area. The Maryland and Virginia prosecutors reached an agreement to try them in Virginia first, and second in Maryland.