The climb up to Hyalite reservoir is nice, about 10 miles with about 1800 feet of elevation gain.
A loop to Kelly canyon. Bridger canyon towards Livingstone (long way 100 miles, via Jackson Creek road and back about 60 miles). Nice long straight decent on the frontage road towards Livingstone, but on the way back it's uphill and usually with a headwind. These are all road rides, plenty of mountainbike trails around too obviously.
Don't know about rentals, never rented one when I lived in Bozeman, and as a poor student did most servicing myself or with the coach of university tri club.

Would recommend: Bangtail bikes in Bozeman and Big Sky Bikes in Missoula

Missoula Bike Works has MTB or road bikes.
Big Sky can get you set up with a really nice MTB. I can't remember if they rent road bikes. Their MTB fleet is usually fancier than MBW's. Both have nice folks working there.

If you're in Missoula on Wednesday night, you can jump in the local MTB race. Look up MTB Missoula.
Bozeman: Bangtail and Owenhouse should work. I've had athletes use both successfully.

What kind of ride do you want? Distance, terrain, road surface?

I doubt I will be doing any mountain bike racing. I am a flat lander training for the hilly IM Lake Placid and the Triple Bypass in Colorado. I could do some mountain bike rides if they weren't too technical. Otherwise I am just looking for some good scenic rides with climbing.

Riding in Missoula for sure Tuesday and possibly Wednesday. Looking for 2-4 hour rides.

Probably in Bozeman Wednesday night and Thursday. 1-3 hour rides.

Sent you a DM.
Have a fun trip!