Howdy Tuftus
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Think this post is a bit out of date but it would be interesting to know if you did take up shooting as a relaxing pass time? I enjoy it and certain disciplines do relax me, mainly target shooting and Clay shooting, clay because the club I go to is set in a beautiful forest which is quite peaceful (apart from all the shotguns being fired).
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I suppose it depends on the type of person you are, if you get frustrated with yourself when you cannot do something (like hit a shot you have made many times before or pick up the skills as fast as you want) then it might not be the best but I think for a lot of people Target and Clay shooting is very relaxing.
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On the flip side of the coin, Tactical shooting I find not relaxing because it’s a scored and timed event, basically it’s a competition and because its time based it can be very intense, so I would stay away from tactical shooting, unless you can purely look at it as a bit of fun and are not that competitive.
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I think if you can find a target or clay shooting club which is outside and not that busy then it can be a very relaxing sport.
Also your budget is fine, £800 would get you a very decent air rifle and pistol and enough pellets to last a while, even for target shooting with live ammunition that budget would still surface, Club membership avg £80, FAC application £50, Ruger 10/22 brand new for max £400 and 22lr ammunition is still quite cheap (about £8-£10 for 100rnds).
Hope this helps, see you on the range