Lots of fantastic suggestions already, I'll add a suggestion that is quite different.
Ruger 77/357 bolt action rifle in .357 Magnum.
Cost-wise, it cannot be purchased as cheaply as many of the entry level bolt guns, I'll admit that right away. But there is a lot of FUN to be had, consider all of these reasons:
--it is short and light, that means it feels like handling your 10/22, it is the epitome of "handy!" and it is the exact opposite of heavy, bloated, gangly and unwieldy. Whether you want to try (difficult!) standing/offhang shooting, prone or the typical fare at a gun range (rested from a bench), this rifle is enjoyable to handle. This size and weight also adds to the pleasure of hauling it to & from wherever you shoot and the short barrel even makes cleaning more enjoyable
--has sights but is better with an optic and super simple and easy to add one with included rings that mount directly to the rifle. You need only choose a scope, all the rest of the hardware is ready and included
--factory ammo is absolutely easy to find everywhere and there is extreme versatility in using the lightest .38 Special possible AND the hardest-core .357 Magnum available and every point along the spectrum in between those two
--if you handload, EVEN BETTER! If you don't handload but you have always wanted to learn... okay, get ready for reckless use of a superlative! (discussion forum post police love to attack superlatives because they are typically easy to pick apart) but listen -- there may not exist a BETTER or safer or more stout platform for using/testing handloads ever invented than a 77/357 rifle.
--also for handloading or learning to handload... .38/.357 is sooooo much more enjoyable at the load bench than any/every bottle neck rifle round ever invented, ever. Bullet selection is huge and low cost, less money spent on powder, published loads available almost to infinity, preparing the brass is light years more enjoyable too. In every facet, .38/.357 is more enjoyable to handload than bottle neck rifle in any caliber
--you said that you would have more fun with a little leas or manageable recoil, the 77/357 has that area cornered! Even heavy .357 from mine isn't at all nasty on the shoulder like some of the .30-cal and beyond high pressure bottle neck rifles.
--your intended range is 0-200 yards? PERFECT for this rifle! Don't ask it to do things well beyond 250/300, it's not a player for longer range. But for 100 yard steel, this rifle is a riot
You could make serious use of this rifle for numerous hunting applications down the road even if you only buy it for FUN right now. And it likely costs more than many of the fantastic package deals they offer on budget combo rifle/scope setups in .223, .243 and .308 but the 77/357 really operates in a different place than those rifles. You might even consider the 77/44 instead, but you will deal with more recoil on the top end.