I'll add more later, too.
Lone Star is worth going to. To get to Lone Star from the Upper Geyser Basin, drive towards West Thumb. Just past Keppler Cascades, you'll see the sign for Lone Star. Turn in and park, then take a lovely ride/walk down the old road to Lone Star. It's considered back country, so there's not a lot of trails and no boardwalks. Make sure that if you do go to back country areas to pay attention because there are hot places that might not be noticeable and a thermal burn miles away from anything wouldn't be fun.
Lone Star erupts quite reliably every three hours. You can ask at the Old Faithful visitor center for the most recent visitor report of an eruption and figure out when you need to head out there so you don't have to wait too very long. It's a little less than 3 miles out to the geyser. There is a river at the geyser that might entertain you while you're waiting. Lone Star usually has a minor eruption about 30 minutes before the major. It lasts a few minutes, then you wait for the major that last about 30 minutes.
We like Firehole Lake Drive. The first neat thing you get to there is Great Fountain Geyser. There is a sign there that might have a prediction of the next eruption. You can also ask at OFVC to see if they have an estimate for an eruption. Firehole Lake Drive will probably have just opened before you get there, so Great Fountain's patterns might not be known, but often it erupts in the 9-16 hour range (12-14 might be closer). It's a fun geyser to watch. One indication that it's not too far from erupting is that water starts overflowing from the crater of the geyser onto the platform. It's usually 1-2 hours from that point, so you can go see something else and then go back to Great Fountain. You can also see the geyser from the parking lot, which is handy in the rain. If you do choose to sit there, you can see White Dome across the way. We like White Dome a lot, but it's not predictable, so it's not really worth sitting next to it even though it's neat.
After those geysers on Firehole Lake Drive, you come to Pink Cone. Ask for a visitor report on this on too at OFVC, since it's fairly consistent, although I don't know what it's doing this year. It seems to be around 18 hoursish?
At any and all geysers, if you see someone there with a notebook, or a backpack, or a walkie-talkie, or talking to someone else about geysers, feel free to ask them questions. There are always lots of geyser people around in the summer. Some of them are testy and gruff, but you can figure that out quickly and leave them alone. Others are delightful and helpful and will show you interesting things, or tell you what to watch for.
More geyser stuff, but the links might be old
In the Lower Geyser Basin, opposite the exit from Firehole Lake Drive, are the Fountain Paint Pots and geysers. The mudpots are always fun to watch, but my favorite here is Fountain Geyser. It usually erupts every 6 hours (ask at OFVC) and lasts for about 30-35 minutes. I love to sit there and watch it. All the geysers in the area are connected and geyser gazers can give you an estimate of whether Fountain is likely to go soon. Jet, which is behind you when you're looking at Fountain, is lots of fun to watch. It goes for short bits often.
These are all things you have a good chance of seeing. There are many other geysers that go rarely. If you hear about Morning, Giantess, Giant, or Steamboat (and others), find out what people are saying because they're rare.
I still need to do the Upper and Midway and Norris Basins.