I'll only second what rebelgrizz said, that you are going to run into LOTS of snow.
Also, I know there's been a great deal of discussion/controversy on hand guns in National Parks, but most experienced back-country folks (i.e., not politicians or the NRA, which are not necessarily the best judges of this sort of thing) feel quite strongly that bear spray is better, safety-wise, for a variety of reasons. Rebelgrizz is correct: carry bear spray (and know how to use it). I believe you currently have the legal right to carry firearms (unloaded, I think), but just because that right is there, doesn't mean it's the best/safest thing to do.
GNP this early in the year is normally a pretty wild place. Be prepared for the potential of quite bad weather, then hope for the best. It has been an unusually mild winter in this part of the Northern Rockies, but the spring can still turn on you.