When i was a youngster, my father bought a .22LR AMT Lightning for me to target practice with. It was basically a knock-off of the Ruger Mark II. This was back in maybe 1982 or so; the pistol is still in my safe (though i need to do some minor work on it now). Get the Ruger Mark whichever you can find.
If you want something less conventional, you might want to check out one of those Ruger 10-22 Chargers; it is like a cut-down 10-22 rifle. As far as i know, the 25 round banana mags for the 10-22 rifle will fit it and be fun for can killing. http://www.budsgunshop.com/catalog/product_info.php/cPath/74/products_id/55794/Handguns/Ruger/Ruger+Rimfire+Pistols/Ruger%2022LR%20CHARGER%2010IN%20W/%20Fired%20Shell%20Casing/
I wouldn't completely writeoff a revolver though. There are some fine .22lr revolvers out there, though many will exceed $300. I personally feel this might be a safer way to get started in shooting depending on the experience level of those who will be shooting with you.
Also, if there is a black powder enthusiast in your family, you might find a black powder pistol interesting. I have a couple and really enjoy them for target practice, though i don't know what i could hunt with them. I have one like this: http://www.cabelas.com/product/Home/Pietta-Revolvers-with-Starters-Kits%7C/pc/105625080/c/112132080/Pietta-Model-1858-New-Army-44-Caliber-Revolver-with-Starter-Kit/731907.uts?destination=%2Fbrowse.cmd%3FcategoryId% 3D112132080%26WTz_l%3DUnknown%253Bcat104792580%253 Bcat104701680%253BMCcat104503680&WTz_l=Unknown%3Bc at104792580%3Bcat104701680%3BMCcat104503680%3Bcat1 12132080
and another that is a single-shot, side-lock, percussion pistol similar to this one http://www.cabelas.com/product/Shooting/Black-Powder/Pistols-Revolvers%7C/pc/104792580/c/104701680/sc/104503680/Traditions-Trapper-50-Caliber-Percussion-Pistol/740925.uts?destination=%2Fcatalog%2Fbrowse%2Fshoot ing-black-powder-pistols-revolvers%2F_%2FN-1100201%2FNs-CATEGORY_SEQ_104503680%3FWTz_l%3DSBC%253Bcat104792 580%253Bcat104701680&WTz_l=SBC%3Bcat104792580%3Bca t104701680%3Bcat104503680
just less fancy. The single shot is one of my favorites for range time; everybody i shoot with wants to try it.
edit* For the record, i LOVE the bersa thunder .380, but not for a first handgun or a range gun. I had one that never jammed and was suprisingly accurate for a small handgun; my sister now carries it. It is designed to be concealable more so than a range gun.