Welcome and congrats!
If the shaking stops after 10 mins, and if the frequency of the vibrations is proportional to the speed, I'd have to diagnose flat-spotting on the tires. Especially in winter, when you park warm tires onto cold cement, the part that touches the surface temporarily memorizes the flat shape. The patch will be larger with underinflated tires. This also gets amplified in winter -- as the air in the tires cools after a drive, the air pressure decreases and the tires deflate slightly, thus increasing the flat contact patch.
Solution?
1. Proper air pressure.
2. Rubber mats in the garage
3. The Tire Cradle (
www.tirecradle.com)... I haven't used them and I find them overpriced, but have got good reviews.
4. Push car a few feet every few weeks if parked for a long time.
Hope this helps.
Z.