Before or after Yellowstone, definitely stay next door at Grand Teton Nat'l Park. Both parks are completely different in things to do and looks. Grand Teton has some campgrounds that don't accept reservations but they each have 300 very nice sites. We've always gotten a site for our 40' motorhome. With your small trailer you'll have no problem. Take the gentle float trip in the park down the Snake River and you might see a moose in the water with you. We did!
If you aren't on any schedule (except for your Yellowstone reservation) explore the surrounding states. There is so much to see and do that one week won't be nearly enough time. You won't have any problems with the roads in the area with your small RV so have a great time!
Each national park has a wonderful web site with lots of good information on things to do, camping and road info. Check them out.
Here are Yellowstone's and Grand Teton to get you started:
https://www.nps.gov/yell/planyourvisit/index.htmhttps://www.nps.gov/grte/index.htm
