For your first gun, especially since this is intended primarily as a plinker, I'd look at something chambered in .22 Long Rifle (22 LR). 22 LR is by far the most affordable ammo you can buy. The cheapest 9mm ammo I've seen is at Wal-Mart, and costs about $22 or $23 for 100 rounds. 550 rounds of 22LR will cost you $15.
Revolver or auto, doesn't matter. 22s are fun to shoot, and will allow you to focus on the fundamentals of stance, grip, breathing, sight picture and trigger manipulation without having to contend with recoil.
In semi autos, the most popular is the Ruger Mark series (Mk I, II and III). My sister bought a very cool blued 5.5" Mk II with adjustable sights recently for $199. Most used Rugers are in the upper $200 range these days, so they certainly fit your budget. Another affordable option is the S&W 22A, though I rarely see them where I live, but I think NIB examples sell for under $400. My personal favorite from a grip-to-hand fitment perspective is the Browning Buckmark. I've seen NIB examples for $300 every now and then, but I've never gotten around to buying one. All are highly accurate and will last you a lifetime with proper care.
On the other hand, with some autos you can buy a 22LR conversion kit which will allow you to shoot say, .45 ACP or 9mm and 22 LR out of the same gun. The CZ-75 in 9mm is probably one of the more common guns with 22 LR conversion kits available, but there are many others that I just can't think of at the moment.
On the revolver side, you've got Double Action (can be fired by cocking the hammer or simply pulling the trigger) and Single Action options (can ONLY be fired by cocking the hammer manually).
In the SA revolver arena, you have the mighty fine Ruger Single Six, starting at around $300. Most folks love 'em, as evidenced by the fact that on average (in my area), used Single Sixes sell for at least as much as NIB examples, when you can find them. Often, these guns come with an extra cylinder that will allow you to fire the flatter shooting 22 Magnum in addition to 22 LR. These guns have a repuation for exceptional accuracy, and if you take care of them, will probably last for generations.
I bought a Heritage Rough Rider Combo myself (22LR/22 Mag), which is basically similar in concept to the Single Six, but it not as nice/more affordable. I paid $150 NIB for mine. The two Heritage SA revolvers I have owned have proven to be mechanically flawless, if not as finely crafted as a Ruger. And to be fair, from my own observations Ruger makes a smaller percentage of lemons that Heritage. The accuracy of both my Heritages has been at least as good as the Rugers.
In Double Action revolvers, on the more affordable end of the spectrum, Harrington and Richardson (H&R) made a lot of cool, affordable DA and SA 22 Revolvers (the 9-shot model 922 being most common in my area), many of which can be found for under $200. They aren't as finely made or as graceful as a S&W or Colt, but they are reliable and plenty accurate. Taurus has the Model 94 (low $300s) and the 970 (high $300s). Suffice to say, there seems to be a rather high lemon rate among the 94s, which I've discovered from personal experience, so I recommend (1) checking them out VERY carefully before buying or (2) going with the larger, slighltly pricer 970, which appears to be of better quality. Ruger sold a small number of 22 LR chambered SP101s (no longer made). From what I have heard, they are nice shooting guns, if you find one. And particularly desireable if you ever want an SP101 chambered in something like .357 Magnum. The last one I saw on Gunbroker (about a year ago) sold for around $425, but I'd bet their selling in the high $400s now.
The gold standard in the 22LR DA revolver market are the S&W Models 617, 17 and 18 (the 17s and 18s also known as the K-22 Masterpiece) and the Colt Diamondbacks. All are super nice, and fairly pricey (particularly the Diamonbacks and pre-war K-22s, aka the Outdoorsman), but worth it IMO. S&W discontinued production of the 6-shot 17 and 18 years ago, though they have re-introduced the 17 as part of their "classic line" recently. NIB 617s (6 or 10 shot) run in the high 600s, and my gut tells me the "classic line" 17s will sell for 700-ish. Used versions of the 617 sell for $100-$200 less. The 17s and 18s usually start around $400 and work their way up. I own a S&W K-22 / Model 17 with a 6" barrel that was made in 1959, and it is one of my favorite guns. I shoot it on nearly every range trip, I've put many thousands of rounds through her and I enjoyed every one of them. It is far more accurate than I. Though they were once common, they are getting noticeably rarer these days. I personally prefer the older 17s and 18s to the newer 617s, as the 617s have an underlug running the full length of the barrel which makes them too muzzle heavy and clumsy feeling for a .22, IMHO.
Good luck, and welcome again to THR!