Great time of the year for the trip. Been there the last 2 years and last year was huge for snow depths. I'm sure there's other opinions out there, but this is my take on the area. Grab a trail map from any of several places in town and you're set.
West Yellowstone - Get a bison burger at the 3 Bear lodge, you won't regret it. Trails run out of town in every direction. Going East is out unless you want to go into the park pay for a guide (required). Worth doing once in a lifetime for sure.
Big Sky area N of town - just countless areas to ride, hills, bowls, it's got it all. If you're adventurous, you can find some of the best kept secret riding areas. 1/2 the trail is groomed, 1/2 is not. Go for the ungroomed and you'll find pure heaven on earth.
Lionhead area - some of the best off-trail deep pow riding within 100 miles. Only real avalanche danger in the area though. Be careful and live to ride another day.
Two Top - Snow ghosts are a must see on any trip to W. Yellowstone. Several routes to get there, all are scenic. Plenty of off-trail riding that is great to cut your teeth on how not to get stuck.
Island Park - Centennial and Sawtell trails are awesome rides and both vastly different. Centennial with good snow is miles of fresh pow as far as you can see. Sawtell takes you on a widing trail through trees, very cool ride.
Black Canyon loop - South of Two top and likely the most scenic ride in the area. Something about riding through those mkiles and miles burnt trees makes you feel like you're on another planet.
You'll not regret this trip, not at that time of the year. Ironically I'm doing the opposite of you and heading to the Snowy's for the first time.