Car rental is expensive and difficult.
If I was making my first trip to Japan (or second or third for that matter), I'd go to Niseko. It almost always has good snow and more people speak English...albeit with an odd Aussie accent
Most of the movies shoot here. It is epic. There's another smaller, more remote place that sometimes works better but you don't want to go there for your first trip. Might overdose you on the Japanese experience ;-)
You fly into the airport in Hokkaido. Take a bus to your hotel/hostel/etc. Ride until your legs turn to jello. Return. If budget permitted, I would also hire a guide for a day or two just so you find the good stuff. I had friends but may still use a guide next time just to find new good stuff.
Bring your own gear. It's expensive to rent things here, sizes are difficult, and I'm not impressed with rental quality (generally here or anywhere). Inexpensive delivery services (Black Cat and others) can take your gear from the airport to the hotel. Your hotel can make recommendations.
After the first trip, you might consider driving but most people don't. Car rental, tolls, driving on the left, etc make it more difficult. Public transportation works so well, there's no need.
Most of Japan works best carless unless you live here and have a place to park. Living on a base, parking is easy for me. Most Japanese folks don't have cars.
Use the site. Google Niseko. You'll also see Annapurna. They're side-by-side. You'll ride both. As with any place, the longer you stay, the better your chance of an epic trip. Definitely you should stay a week or more just because travel is so expensive and time consuming. You want to get your money back.
If you don't have one, get your passport in the works. It's not always a quick process and you can't go without it.