Airgunner,
Of the three choices you stated, Beeman GS1000, the Gamo hunter 220, or the Winchester 1000x, the worst of the lot is by far the Gamo 220. It isn't half the rifle that the Gamo 890 is. I have not heard any good remarks about the 220. It is something you buy now and then replace in a year or so due to one or more failures. My daughter bought a Gamo 880(before they changed the number to 890) and I just replaced it with a Winchester 1000X because of it's loosing pressure. Seals failed. The Winchester is a great value at the under $80.00 price tag. They are reliable, accurate but the only knock on them seems to be the heavy trigger pull. At 5 lbs.(+/-) it is something you can get used to. I know a few people and a couple of poster here that have the Winchester and they have been great shooters. The Beeman is, of course, the top of the line of the ones you asked about but you are going to pay more for it. As far as caliber goes the .22 is a better hunting caliber but I've taken many, MANY Jack Rabbits with the .177 cleanly. All depends on your choice of pellet. If you're wanting an all around caliber, target and hunting, then between the two I would pick the .177. Lawdog