I would go with the 10/22 as a first .22. As others mentioned, they are easily customizable to make them a real tack-driver. They are available at just about anywhere rifles are sold. My local Big 5 had the standard carbine on sale last week for $179. Wal-Marts will have them for about the same price everyday but I don't know if CA Wal-Marts still sell guns or not.
A semi-auto rifle is great for plinking out in the woods but you go thru ammo quickly so pick up an extra mag or two. 10/22 mags can be changed quickly but the standard mag release is a bit small if you have large fingers.
Add a scope for the outdoor days for more accurate shooting. 10/22's are drilled and tapped for a mount. Use a Weaver TO-9 rail (about $6.00) and get this Bushnell scope at Midway USA:
http://www.midwayusa.com/rewriteaproduct/887219 It comes with rings that will work with the Weaver rail.
My 10/22 seems to like most all of the bulk pack LR ammo but the Winchester DynaPoints from K-Mart seem to be the most accurate. It's more expensive ($12.99) compared to the Remington Golden Bullet at $8.99 at Wally World.
The Rimfire Central Forum (
www.rimfirecentral.com ) has 4 different sections devoted to the 10/22 rifle. You'll find a tremendous amount of information there about what you can do to the rifle and what it is capable of.
I would also suggest a .22 pistol at some point after you have had some fun with the rifle. I have a Ruger Single Six revolver that has both a .22 long rifle cylinder and a .22 WMR magnum cylinder. This is a really fun single action revolver that you should be able to find in the $320 range and will give you years of fun. Mine is 28 years old and always goes to the range no matter what elise I take.
A good choice for a pistol is either a Ruger MKII or a 22/45l. The blued standard barrel versions are around $250 or less. Most of the bull (heavy target barrel) barrel versions go for around $300-$375. This link to my local gun shop will give you an idea of prices and the varieties available:
http://www.eastcountyguns.com/rfrugerpistols.htm
Of course having both is the best way to enjoy cheap shooting!!