I'm 6' 4" and 220 pounds (ish
) and I prefer lighter guns by a country mile.
My current favourite and mainstay is a Venom Stage 1 Lazaglide Weihrauch HW95K break barrel in .177
Even an off the peg Weihrauch HW95K is a stunning bit of engineering - they have superb trigger units, they are bullet proof (no pun intended) and mine has had thousands upon thousands of shots through it and it is incredibly consistent and easy to handle.
I have far more expensive and supposedly 'better' guns but this is the one I reach for again and again above the others. A mate of mine recently bought a Weihrauch HW99S which is another great lightweight contender.
Air Arms are superbly engineered but I got rid of mine purely because of the weight. I do a lot of miles when shooting and I don't much enjoy lugging a heavy gun around with me, especially after a successful session where I might have to make two or three trips back to collect all the rabbits.
Find a club or, at the very least, a shop in your area and go and try some guns out, even if you only get to shoulder them. Often you will go in with your heart set on one product only to completely change your mind once you shoulder a gun that is particularly comfortable.
Keep in mind that some of your budget should go towards a scope at some point, although it is feasible to buy a gun and learn to shoot and hunt using the open sights - often this is a good way forward because your effective range will be less than with a scope and so your fieldcraft has to be well above average to get within range. It's a useful part of the learning curve.
Guns change hands often and there is usually no shortage, but stay well clear of anything where the stock bolt heads show signs of chewed up slots where folks have done a sloppy DIY strip. Guns that have been taken apart with no regard for using the right tool are unlikely to have had their inner working treated any better, so look for witness marks on the stock bolt heads and use their overall condition as a starting point to indicate whether you should bother to look more closely at the rest of the gun.
You don't have to spend fortunes to get great results but you do have to spend the time to get the best from your kit. Even modest gear can normally outperform the talents of the person shooting it, so don't focus on a particular brand too hard - concentrate more on that vital 'click' when you shoulder the gun. Bottom line - if it feels comfortable when you shoulder it you're going to be a better shot with that gun.
Putting the time into learning your fieldcraft and stalking, practising shooting in various conditions and in different positions (prone, kneeling, standing, shooting up or downhill, into, with and across the wind and so on) are what makes you a successful hunter. A decent gun helps, but if you get the fieldcraft and technique wrong it really doesn't matter how much you spent on your gun or what brand it is...
Finally, I'm not going to get into a debate on calibres other than to say don't get too hung up on it. Hitting what you aim at is more important than the calibre you use so if you find a gun you are comfortable with and the power has been tested to be comfortably within the legal limit, go with it. Both mainstream calibres have various merits and they also have failings - if you go for a .22 you could do worse than check out some of the latest pellets to be released which fly a lot flatter than normal. You pay more for them but they offer a more level and predictable flight trajectory and reduce the amount of hold over/under required at long/short range. If you buy .177 you can almost aim dead on at anything from 15 to 35 yards and you won't be more than an inch out anywhere in between. Both can do the business, but only if you take care of yours.
I wish you the very best of luck - airgun hunting is a lot of fun and offers challenges that set it well apart from most other forms of hunting.
