Bear Canyon
From Bozeman: Take I-90 East. Take Exit 313. Right on Bozeman Trail Rd. Left on Bear Canyon Rd. Follow to end.
After the end of the road at the BLM gate, you will need to hike about ½ mile to get to a set of cliffs across the creek.
There is no camping here and the access route is steep.
Bozeman Pass/Rocky Canyon
Take "Trail Creek" exit on I-90 and head North of interstate until first left is encountered. Drive to end of road past pink house and park at new small lot. Walk west for .8 miles along trail/roadcut to "old" parking lot and trail.
This is a fun limestone climb located right on the Interstate.
Gallatin Canyon
20 miles from Bozeman on US 191
The limestone face gets better the farther you go into the Canyon. You will need at least two ropes and a good selection of mixed gear.
There is lots of camping in the area, so you can easily make it a multi-day trip.
Hyalite Canyon
10 miles outside of Bozeman
This canyon is a popular destination for trad climbers as it has several crack, face, and chimney climbs.
This can also be a multi-day excursion, as there is lots of camping in the area.
Madison Range
Various access points within close driving distance from Bozeman.
A big, big mountain range. Lots of technical rock, great ice routes.
Many free places to camp.
Both Bear Mountain and Bozeman Pass are really legit and well bolted/maintained. There is also some climbing out near Livingston that I forget the name of that also has some really fun routes.
There are these massive fake Bouldering rocks around town too that are really fun for a nice quick climbing sesh.
http://bozemanboulders.blogspot.com/ Has some good info about them.