Medora is the town right outside the South Unit of the park. It's a tourist trap, but it's worth walking around grabbing lunch or an ice cream cone and seeing the sights. You might also consider staying the night there. Depending on the time of year it can get very busy and finding a room might be difficult. This can be true for some of the camp sites too.
If you know your timeline, I'd book reservations ahead of time. Throughout the summer there are several different events going on in the town of Medora itself. Hot air balloon festival, rodeo, covert rally, etc. These can add another thing to do, but also tend to draw an even bigger crowd.
Outside the park itself, the Medora Musical is the main touristy attraction. It's worth seeing once. They usually bring in some other "acts" to break up the show and provide a little variety. Prior to the musical, they also do a "Pitchfork Fondue". It's a buffet style dinner where the gimmick is they skewer steaks onto a pitchfork and then dip it into a large vat of boiling oil. It's a little overpriced for the quality of the meal, but you're really paying for the experience. Tickets for the Fondue are sold separately from the Musical and you can attend either without attending the other, but they are billed as complementary experiences.
There are tons of trails for hiking, biking, or horse riding. I believe there are couple spots in Medora that you can rent bikes and the like if you don't want to bring your own. There is also guided horse tours of a couple different areas of the park that you can sign up and pay for if that interests you.
If you're family enjoy's golfing Bully Pulpit is considered one of the most beautiful (and challenging) courses in the region. It's pretty expensive and will likely require you to book a tee time in advance if it's something you want to do. Be prepared to fight the wind though.
I'd also endorse the North Unit of the park. It's about another 1.5 North of Medora, but it's usually less crowded. I personally think it has better trails for hiking. I also find the park rangers to be a little more relaxed and willing to talk.
They generally keep track of the different animals and if you ask they'll point you in the right direction to try to see them.
You're pretty much guaranteed to see at least a couple bison (although probably not a large herd like in the South Unit), and it's the only part of the park that I've been able to catch a glimpse of a bighorn sheep.
If you do decide to go up to the North Unit consider going a littler further to Watford City to Stonehome Brewing. It has a nice selection of beers and pretty decent food.
You'll also be near Lake Sakakawea (ND's largest lake, albeit man made) which is the resulting reservoir from the Garrison Dam on the Missouri River. Garrison Dam is neat, but probably only to fill a couple hours. They do a tour, but you'd want to check times and possibly book ahead. I don't believe they let people just wander through unattended anymore due to the potential security risks.
If you're willing to travel East to ND's capital Bismarck, there are a couple other things you could consider checking out.
Others have mentioned the Enchanted Highway. Yes, the large metal sculptures/displays are impressive to see in person. However, after about the 3rd one it starts to lose it's novelty rather quick.
Fort Lincoln State Park just outside of Mandan on the other side of the river from Bismark. Overlooking the Missouri River, a lot of the Fort is still in good condition and they have employees dressed for the era giving tours and acting as if you're back in the time when the fort was active. I highly recommend timing your visit to participate in a guided tour. The "actors" are great and really know a lot of facts and stories about the Fort. Once the formal tour is complete, you can explore or revisit at your leisure. Consider packing a lunch and having a picnic as the food and beverage options are very limited within the park itself. Honestly, since you seem to like the National parks this would probably be my top recommendation, as it's a very similar vein to some of the other attractions you listed.
I'd also recommend stopping by the capital building. I believe you can schedule a tour of the capital building itself, but there is now also an attached museum that has some pretty neat exhibits and really does a good job of explaining some of the more important history of the state.
