I was where you were 10 years ago. I started out with a CZ 455 American (for use with a scope) and a Marlin (for use with open sights). I guess if I were going to start now, I'd take a slightly different approach. There are a couple of things that I have found out about my own shooting.
1) The quality of your barrel is, in my opinion, the most important single piece of this equation. All factory barrels are a bit of crap shoot -- you get what you get, and half of them are, definitionally, below average. So if accuracy is your goal, one consideration is whether your platform can be upgraded easily. CZ fits the bill here. The Ruger 10/22 does too. The barrels on these can be swapped by the shooter with no gunsmithing skills for barrels made by modern masters. So can the triggers. So consider the pathway rather than simply the entry level rifle. I did have a Ruger American Rimfire and you couldn't easily swap the barrel on the thing.
2) Stock shape is almost as important to accurate shooting as the barrel. Consider that for five shots to make a ragged hole in paper at 50 yards, everything has to be nearly the same from shot to shot. This includes rifle placement on your rest/support. Sporters are a real challenge for this, because the fore-ends are rounded and hard to control for cant, and the butt-stock is angled sharply to the wrist, causing the muzzle to change elevation with every shot. I am not saying you can't put five shots in the circumference of a dime with that Anschutz pictured above, but your work will be easier with a stock that has a flat foreend and a buttstock that is parallel to your shooting surface.
3) I agree with those that say that a CZ can be made to shoot with an Anschutz . . . and that it will cost money to get you there. I have done this a couple of times and it has required: barrel swaps (Lilja $500), new triggers (around $250), bedding jobs to fix stock fitment issues and take action-screw torque out of the equation. All in all, I might have been better off just buying a used Anschutz, because that's the price territory I got into with all my modifications.
4) Save some money in your budget for a scope. 14x minimum at the high end. 24x is much better. Think used -- search the classifieds here.
5) Everyone has their own rules of thumb about these things. I think of the ratios as 1/3-1/3-1/3: shooter, rifle and ammo. A great rifle will never shoot inconsistent ammo into small, repeatable groups even if it does occasionally commit a random act of accuracy. And you can spend all the money you like (I did) and unless you master the basics you won't be able to buy a bullseye (at least, I couldn't). You may laugh at this -- but the reason you shouldn't sweat the buy-in price for your rifle is that you are likely to spend a lot more money on ammunition over the rifle's life than the price of the rifle itself.
6) Don't forget to have fun.