Of course there are lots of mountains in Wyoming and Montana outside the national parks, just as spectacular and a lot more wild. I go there, not to either Yellowstone or the Tetons. It's just that the OP was planning to stick to national parks.

Yellowstone is basically a forested plateau of about 8-9,000 feet elevation with not much scenery to look at, especially if, like me, your idea of scenery is rugged mountains. I personally can't stand the stinky hot water, either! Yellowstone's only saving grace for me is the canyon (crowded) and the wildlife. With the lodgepole pine forests now dying from the bark beetle epidemic, I'd rather be up around timberline!

Sticking to established trails close to the front country (where people are more apt to come along should you get hurt) is a good plan for solo hiking. Especially be more alert for risks. Wandering through boulder fields, rock climbing or other risky off-trail activities are not a good idea when you're solo. In other words, if you must break a leg, it's best to do it when someone else is nearby! As already mentioned, letting a reliable relative or friend know your itinerary and schedule is extremely important, too. Be careful about using the "911" button on the SPOT; you don't want to do that unless it's a true life-and-death emergency!

Since you say you haven't done a lot of backpacking, I'd suggest making your first few trips really short ones, relatively close to the trailhead so you can bail out if needed. Practice going out in bad weather, so you learn to stay dry. In the Rockies, you can expect thundershowers to show up most afternoons. It's a good idea to start early in the day and be off the trail by early afternoon when this activity (lightning is the biggest hazard) is going on. Fortunately, they normally don't last long.