I've been to Breck many times, every time with a car. That said, a car is really not a necessity in Breck as the public trans is outstanding! You can get around town via the "trolley", (town, grocery store, etc), and into Frisco, Dillon, Keystone and even Copper Mt. via Summit Stage; all these are free. Most likely discount lift tickets will not be available in Breck, but they should be in Frisco. I used to get mine at the Saveway there, near the Antlers shop mentioned above.
If you want to go for a day to Vail, you can get a bus, (pay, but not much), from Breck. It leaves from two locations every morning. You'll need reservations for this, and in the past, I called the Keystone Activity Center for this. The bus drops you at either Lion's Head or Vail Village. I'm pretty sure you'll have a hard time getting a discount on a single day at Vail, as they stopped that a couple of years ago if I remember correctly. Multi-day tickets from Vail Corp., (Vail, BC, Breck, Keystone), are usually good for at least one day at Vail, depending on the days purchased.
I don't know if this is your first trip to Summit County, but if it is, you have picked a good spot. I've never been disappointed!