No, those break opens ... aren't pumped. Break open the action load 'n go. You won't be doin' any stingin' ...you'll be doin' killin'.
Good place to start is Air Gun Depot (http://www.airgundepot.com). You can get some power .. for a little plinker. I recommend that you stick with.177 .. But, that's personal preference.
You have a garden and you don't want to shoot rabbits?
You will have a blast. Of course .. there is always a downside. You'll start out with spending around two hundy .. the next thing that you'll get into is more power .. then, you'll wind up with those semi-auto rifles ... customize them ... and then ..you'll start dropping a couple a hundy, every week on ammunition and range time.
Then, you'll get into REAL rifles, like Weatherbys and Kimbers .. Then, you'll start wanting classic German and English pieces ... Try explaining a six figure purchase for an English Double rifle to your better half. Them wimmin kin git downright mean when theys upset..
Oh it's a twisted web that you shall be entering ...
Or, you can be a sane and sensible human being and not become a fanatic. But ... where's the fun in that. Remember, you just can't put a price on fun.
I, personally, never fell into the AR type or Kalashnikov type trap. However, I did not escape the Weatherby/Kimber/Classic firearms route. Oh ... It's terrible to need a fix.