always, always, always go get the battery checked at one of the places that do it for free. autozone, checker, oreilly, if nothing else.
if you can start your truck, unhook the battery and it keeps running, then it is the battery. if it dies shortly, it is the alternator.
the starter will be a problem if it won't attempt to turn over. i generally recommend changing out the positive lead to the starter before changing it out. 12 bucks max and it will be good for another 15+ years (never dealt with salt so don't quote me on that).
edit: there is no reason to 'drop it off at the dealer' for this, unless you are really uncomfortable with basic troubleshooting.