Big Sky just has a couple of zones off the top of lone peak that require you carry a beacon, shovel, probe and have a partner. These areas aren't always open and if it has recently snowed, probably won't for a couple days, ie the North Summit Snowfield. The Big Couloir is the most likely area like this that will be open and its a pretty wide couloir but the top is pretty step and narrow at first.
There are other areas like the headwaters and a-z chutes, etc that are hike access and will require this equipment. Again, most of these runs aren't always open and they are pretty gnarly for the most part. Unless you are a very experienced skier, probably won't need to worry about bringing any avy gear and not judging you personally (just a generalization) but if you're from the east, you will likely have everything you want in-bounds at Big Sky. There are plenty of super steep, super narrow runs off of Lone Peak and Challenger/Headwater lifts that you will be stoked on and don't have to worry about avy gear.
To OP, Big Sky is a huge mountain with terrain that satisfies all skiers. If you wanna cruise groomers all day, good luck skiing every single one in a day, there so many epic cruising groomers at Big Sky. Of course there is a park, it seems to be doing pretty good these days, only one run I believe. Head up the mountain to the main bowl, lone peak, challenger, headwaters, lone pine to name a few. Upper mountain has plenty of steep terrain, lot of good tree skiing, etc you won't be disappointed.