For at home, a stationery
recumbent bike might work for you. I found an older one for myself on Craigslist for less than $50.
Many exercises can be done while sitting in a chair, or even lying in a bed. Leg and arm raises are a great way to start. Sitting in a chair and marching your feet in place or swinging your arms can get the heart rate going with very little stress on the joints.
Here's an excellent video
Sit & Be Fit to start out with.