I haven't bothered with the passport, but I do try to get a refrigerator magnet at each park I visit.
Yellowstone is so huge that you can't do it justice without at least a few days there. The park has several distinct regions with different stuff to see and do in each. We kind of hit the high points at most of them, but I could easily spend weeks there exploring in a lot more detail. To me the best part is just the sheer amount of wildlife. Bison, elk, antelope, bears, mountain goat, bald eagles, river otters, beavers and I don't even remember what else we saw. We went in early June, so it was perfect spring weather and all the wildflowers were blooming, so hikes through the mountain forests and meadows were gorgeous. Plus there's the largest lake above 7000' in North America. Not to mention the geothermal features. And waterfalls.
And of course, Grand Tetons is right next door and it's awesome too.
You're in a great location for outdoors stuff, with Rainier and Olympic and all the mountains to the east. It sounds like you're not really confident in your outdoor ability, but I bet you could find some outdoor groups that would be happy to show you the ropes. We have a couple outdoor clubs in my area that do a lot of camping and such. One of them had a group that sumitted Rainier a couple years ago. They offer classes in backpacking, mountaineering, winter camping, etc.