so let's break this into two things - driving and sightseeing -
Driving - the driving in either Teton or Yellowstone is never good. The speed limits are very low (45 mph for much of it) and the roads will be crowded during the summer. So forget sporty driving in either of the parks. The Beartooth pass is the real deal for a driver's road but you need to hit it early in the morning before the campers hit the road and try to get over the 10,000' pass. If you go early you only need to worry about wildlife - Elk and muledeer. If you leave Yellowstone and go over the Beartooth you will drop down to the sweet little town of Redlodge. Good food, lodging and shopping there. I'm not sure when the pass opens in the spring but it's late. Be sure to check the projected opening dates before making even rough plans.
There is also a road headed north out of West Yellowstone that goes up to Bozeman and that is route 191. Twisty and quick it can be a lot of fun. It is right next to a river (where the movie "A River runs through it" was filmed) and there is very little passing so slow traffic can ruin the whole thing. Rt 191, or "the canyon" was at one time designated the 2nd most dangerous road in the USA due to trucks rolling over and icy fog. June should be good but it can snow 10 months of the year here.
There are a number of great drivers roads near Bozeman. Most MT roads are open, straight and very quick but not very challenging - that said you can find great roads around Bozeman -
* Bridger canyon road from Bozeman to Wilsaw and Livingston - my favorite with very challenging roads with lots of elevation gain/loss and very curvy.
* Hyalite Canyon
* Jackson creek road
Sight seeing - this is endless. Yellowstone has the best diversity of stuff to see. If you've never seen 'Old faithful' it may be worth the wait and crowds but otherwise there are much better things to see. The madison area is great as the the boiling mud pots and grand prismatic spring. Simply stunning.
The best part of the park IMO is the Lamar river valley which runs east-west in the northern part of the park. With the exception of the road itself you'll drive on it looks like it has for thousands of years and you will have to work hard to not see a lot of wildlife. Drive slow and look around and you can see bison, black bear, grizzly bear, elk, muledeer, wolves, big horn sheep, antelope and even badgers. I've seen all of these in the Lamar.
The Tetons are wonderful too. Different than Yellowstone but HUGE. One needs to drive and take advantage of the pull-outs so that the driver can really look around. The town there is Jackson Hole and it is very touristy - some folks like that and some don't - I don't personally but you can buy lots of expensive stuff like furs and $100 cigars if you like.
The drive north out of Yellowstone toward Livingston will put you in the Paradise Valley and it's aptly named. High mountains all around with stunning views. Chico Hot Springs is there and worth looking into for lodging and very, very good food. Don't go for the cheap lodging option as it is 'rustic' at best. Pay more and stay in one of the upscale rooms and you'll love it. The hot springs feed an olympic sized pool full of hot spring water that bubbles out of the ground and flows through the pool. Really fun. Chico is just south of Livingston and about an hour from Bozeman.
One thing to keep in mind about traveling here - nothing is close to anything else and the shear amount of space and distance can be hard to reconcile if you haven't been here. In other words just because the two towns are close on the map doesn't mean much. If you are in a town you will have everything you need but when you leave town there will be nothing at all. No rest stops or quickie marts - it is open and empty. i know folks that drive 45 minutes to buy a jug of milk. I live in Bozeman so everything is close at hand but plan accordingly.
I hope that helps.