This is the third highest mountain in the Teton range, and a proud peak in it's own right. It hosts a number of routes ranging from scrambles to mixed, ice, and rock routes. The easiest route is the SW coulior, which is mostly snow with third class rock. There are a couple short fourth class steps, but it's not too exposed. Other routes include:
North Ridge 5.7
Northwest Ice Couloir AI3
East Buttress 5.11
Buckingham Ridge 5.7
Middle Teton Glacier 5.7 AI3 For a gripping account of this route, check out this website: larsonweb.com/Bctry/TripReports/Climbing/MiddleTeton.htm
Get to Jackson Hole Wyoming. The Tetons lie just to the North of town. Begin at the Lupine Meadows trailhead, and hike up Garnet Canyon, the same approach for the Exum Ridge on the Grand Teton. Many climbers choose to camp here, and climb the Middle Teton in preparation for the Grand Teton. The view of the Grand from the Middle is, well, Grand. Exum Ridge is the central feature. For those climbing only the Middle Teton, it is possible to climb it in a day by the non-technical routes. If you've chosen a Northern route to climb, it's also possible to camp at the lower saddle between the Grand Teton and the Middle Teton. Exum mountain guides maintain a large hut here for their clients.
Overnight permits are free, but required. There are quotas for the popular backcountry campsites. Pick up your pass at the Jenny Lake backcountry ranger station. The climbing rangers in the Tetons are some of the best in all the National Parks. They give accurate information without advising, and can be very helpful. Day climbs and hikes do not require a pass, but it's a good idea to sign in, especially if your plans are ambitious.
There is also a fee to enter the park, if you're climbing in the USA a lot, the best thing to do is to buy a Golden Eagle Pass, which functions in all National Parks, and in many other federally controlled areas, as well as some state parks.
Those choosing to sneak into the park without paying, or even worse, not paying the camping fees, or worse still, parking illegely, my be fined and or torched.