Cloud Driver is right, going through Idaho is longer but would be, in your case, better than trying to thread through downtown Jackson! (There is a truck route through Jackson in case you do - watch carefully for it. Easier to find southbound than northbound. Definitely preferable when towing the rig over going the regular route thru town.) 26 from Alpine through Swan Valley, Idaho is one of the prettiest drives we've taken (and can be spectacular in the fall); good road, and there's a nice rest area west of Swan Valley - good place to stretch your legs.
Your main climb will be on US 20 on Ashton Hill, just north of Ashton as you head up into Island Park. It's only a few miles. There has been construction on that route this summer; we haven't been there for over a month, so not sure what the road conditions are there now. Dial 511 from home phone or (if you're in that state) from your cellphone and you can get local road conditions with minimal number-crunching - works in the states around here, anyway.
Gas stops: if you do take Cloud Driver's route, fill up in Alpine. There is a gas station there your rig will fit into. Nothing for big rigs after that until you're basically in Idaho Falls. Island Park has a few stations large enough for you - keep an eye out while driving through there. West Yellowstone's gas is A) very expensive and B) very few large gas stations capable of holding your rig (especially with the trailer in tow).
