This is not medical advice.
I get asked this question often. Usually it’s by people being discharged from physical therapy. Many people who have had a total knee replacement should consider a recumbent stationary bike. Some people ask about treadmills, however walking involves joint impact, and very little joint range of motion. This is true of all knee issues. A stationary bike provides muscle strengthening and constant ROM. Joints have fluid and viscosity, and feel better with gentle and constant ROM. This is partially why your joints are so stiff after sitting for a while.
What kind of equipment also depends on what your goals are. If you are dead set against a health club, then some equipment at home would be nice. You don’t want it to become an expensive coat rack, which happens often. It’s best to be honest with yourself: Am I going to use this?
I’m generally not in favor of treadmills because it’s more of the same; we walk all day. The body will respond more to a stimulus that it’s not getting, like bicycle motion, or weight lifting.
If I could only buy one piece, it would be a Total Gym, with the arm cable attachments. There are wildly different models over the years, with some models in the $400 range, up to thousands for clinic models. The lower range is fine for home. I picked up a used one on the internet. It was too cheap at $50, but it does some of the things I was wanting.
The Total Gym uses a percentage of your body weight for resistance, and is generally smooth resistance on rollers. This can be set very low for safe resistance. Some people may have a tough time reaching all positions. There are risers you can buy, or put a strong box under the bottom, to raise the base level. Start the resistance and board very low at first, especially with the arm attachments, to avoid shoulder injury.
Brands
Buying an expensive piece of equipment can be risky, especially these days with supply issues and heavily globalized market. You don’t want to buy too cheap, as this will be a greater hassle trying to fix. Does the company have a reputable name? Schwinn is still around, and makes decent quality equipment that seems to last. Unknown brands may leave you stranded when needing repairs. If you buy an extended warranty, who honestly is going to pack this up, ship it off, or come to your remote part of the world and fix it? There are some more expensive routes that will garauntee delivery and set up.
The stationary bicycles are a good example in differences in quality. Generally you want smooth magnetic resistance, which entails electronics. This can be a source of breakdown. The nice thing about the Schwinn Aerodynes are that they are like road bicycles, and can sometimes be repaired by a local bike shop. They are heavy, but stable. They have smooth wind resistance, and sometimes an arm attachment. Be wary if you are too weak or unbalanced to get on and off the higher seat.
The bikes you don’t want have friction resistance, which is annoying, and you are unlikely to ride.
Rubber bands are a great way to add easy resistance. My favorite brand is Slastix. They have handles, and are enclosed in a cloth tube, so they are more safe if they break. You can put them anywhere, on sturdy door handles and around posts, over beams, in the garage, on the porch, etc.
Exercise balls have a variety of uses, and the internet is full of examples. Gimicks like the thigh masters and shake it weights usually just end up at garage sales. Rowers are nice and can be very smooth, however they only do one thing. You have to really like performing that rowing motion, over and over again.
To get started on lifting weights, check out this link.