Buy a decent mower for your one acre yard and hire out the 20 acres to be mowed.
For the yard, an acre isn't much, but what you buy is really dependent on your budget. I have a Scag Wildcat that cost $8,000 for my nice areas. It does a heck of a job and is ten times the mower my Craftsman riding mower was that I bought used for a grand. I used that Craftsman to mow a half acre yard once a week, then then to do several areas when I moved to where I live now. It worked fine and I was looking around to replace it with another when I went crazy and bought the Scag. You don't need that much mower for a nice residential yard, but it sure is nice.
Buying a tractor and cutter to do 20 acres is going to cost ten to twenty grand. It takes allot of HP to run a bigger mower, or cut tall grass. If you do it once a month, you can get by with a smaller tractor, but then you're also out there allot more often and putting hours on the tractor. Then there is maintenance, repairs and storage.
Figure out what it will cost to buy it and run it and you'll find that hiring it out is going to be allot cheaper.
If you've never sat on a tractor and mowed anythng of any size, in rough condition, you're in for a big suprise there too. It's only fun for about the first hour, then it turns into work and then it becomes uncomfortable, painful work. Once you buy the tractor and cutter, you'll be stuck having to run it all the time, or find somebody else to do it. Most will want to the first time, but then never again.
Again, hire it out and see what happens with that approach.
Good luck,
Eddie