Some very good advice in the previous posts. Crossman makes a fun plinking pistol. For the money, they are hard to beat. I have the 1322.
I would add, for casual backyard plinking, and pest control / hunting, a nice moderate to high powered springer long gun is hard to beat. I have always been a fan of the Diana springers. The 350 is way too loud, too hard to cock repeatedly. I got rid of my 350. The 34 is perfect. Sidelever model is also nice, I still have one, a model 52 in .22.
My all time favorite is the Diana model 45 .177, bought one in 1988 (900 fps with a rubber piston seal), foolishly sold it, then found a replacement, a 1983 model on Ebay a few years ago. This one has the leather piston seal, shoots a bit slower...probably around 800 fps, just needs a drop of oil on occasion to keep the leather alive. You can still find the RWS Diana 45 used if you look.
Properly cared for, springers last indefinitely as long as you don't store it cocked or dry fire it... or loan it out to friends.
Springers are perfect for survivalist minded folk, as there is just FAR less to break over the long haul. No tanks, pumps , valves or hoses! Just a spare spring, an extra breach and piston seal along, bottle of the proper oil with plenty of pellets and you should be good come Hell or high water. Something to consider in today's crazy world.
My vintage Diana 45 is perfect for plinking tin cans, squirrels, rabbits, feral cats, rats, crows, etc. Over 800 FPS in a man sized rifle, German made wood and blue steel, accurate & classy, what more do you need?
Oh, pellets. You need pellets.
I like the RWS pellets, pricey, but accurate. Still far cheaper than .22's. .177 is potent on birds, but with squirrels, rabbits, etc., I would suggest head shots.
The 45 is no longer made, but the 34 is its son if you want brand new and or in .22, which is probably better for hunting / pests. Or, my next likely airgun, a Weihrauch HW50... another quality made air rifle. Very nice gun.
Don't get too hung up on speed. Some people think you have to have the fastest airgun made. Speed sells. You do not need 1,000 fps to shoot rats or tin cans. 650 to 800 is more than enough, and many of us have been successful and satisfied with even less. Shot placement is the secret.
With a springer, use the artillery hold, no death grip... hold it loose, allow it to recoil foward. First lesson I learned in 1988 with my old 45 Diana. Springers do have a learning curve, but once mastered are a true joy to use.