Most of the options mentioned I currently own. I must say that everyone has different ideas of fun and what fits me best may be of no interest to you.
I have become hooked on prone shooting with a sling using an aperture rear sight and a globe front, though a post front is fun too. As such a rifle that fits this shooting goes highest on my list. The Savage Mk II FVT is a great option here. The thing shoots well, the rear sight has been precise and the front globe comes with many inserts and fits standard 18mm Anschutz front inserts. The only problem is that stock. It flexes and it is designed for iron sights that hug the barrel, not sights that sit off the barrel an 1.5". As such I have a terrible time getting consistent cheek weld, or more like chin weld. So if I went with the FVT, which I love and would not get rid of for the price they ask, I would look at either picking a take off wood stock to both increase rigidity and raise the cheek weld. This bumps the price a bit but it is my strong suggestion. The great thing is that you can remove these sights easily, toss on some scope bases, and move to a scoped rifle down the road.
Another aperture option would be to go with a CZ trainer and buy either the old BRNO sights that brnomann on rimfirecentral sells or to buy the rear aperture sight from williams. Both are decent options. The williams sits low and uses the stock front post, so the stock will put your head at the right height. The BRNO sights sit higher and will require either a new stock or a cheek riser to get good cheek weld. These will give you the front globe type sight with inserts which I prefer to being stuck with just a front post.
That brings me to the CZ lineup. I have two, a trainer and a Varmint. Enjoy them both. They are a step up in quality from the Savage line. Mine both shoot a little better than the Savage, but not a ton. I use the trainer to hunt and the varmint for bench shooting. Both fit these uses well. They are a little expensive and for the extra money spent I'm not sure I wouldn't go with the Savage as my only rifle and then 5,000 rounds of ammo for the difference on getting a single rifle. I am not getting rid of either of mine, but they do command a premium price and if budget is a concern, they are going to be pushing it to get the rifle and a scope of any quality for $600, let alone $500.
I have a 10-22 as well. It is an ok rifle. I find people tend to either love them or hate them. I am a bit more neutral. I restocked mine, put on a new barrel, and put in a new hammer to get it where I wanted. It shoots pretty well. Not quite with the Savage MK II but certainly as good as I could expect from a semi auto. It is fun to blast with. With the bull barrel it feels a bit front heavy for off hand shooting. I got lucky in that I found the stock dirt cheap and I got the barrel when Green Mountain was selling packs of seconds so I don't have a total of $300 into my 10-22. Had I paid for a new stock, a new barrel, and the trigger I would have been easily over $400 if not $500 to $600. Things add up quick and I question if I hadn't spent too much already.
The marlin 60 is another quality semi auto. Being the most sold rimfire of all time and having one of the smallest aftermarkets means the factory must be doing something right. I like a tube fed .22 but some don't. Pretty accurate out of the box. Just depends if its what you like.
There are a few more options. You can look at an older Remington or Winchester for that price and get a real nice rifle. The CMP Kimbers are ok. I haven't been too impressed with them. From all accounts I have heard accuracy is nothing spectacular. Not bad, just nothing a $300 Savage doesn't do. I have also heard the reason these rifles were sent to the CMP unused was due to them not holding the accuracy standards the quoted and the contract was canceled mid way through. Don't know this to be more than rumor and I'm sure more than a few of them shoot well.
Also there is the Marlin bolt action .22. It sounds like it shoots well and is inexpensive. There is a Remington semi auto out there as well that gets decent reviews. I can't stand the build of the mags and that killed my interest from the beginning. Don't know how they do.
I'm sure there are more, these are just the ones I am most familiar with.
Really you first need to ask if you want a bolt action or a semi auto. That makes a big difference in the suggestion. Then you need to ask if you are willing to pay a premium for a bit of build quality, fit, and finish. Then we need to know if you have interest in iron sights or not. From there it gets easy to suggest a rifle or two that fit the category well.