I went through
exactly what you went through last year, so perhaps I can give you some useful advice. I may be a thourough (long winded in other words.

) But hopefully it will help you out.
I was just getting into handguns early last year, and I was asking the same question... dedicated rimfire pistol or conversion?
I ended up picking up both a Ruger MKIII and a Browning Buckmark.
Ruger MKIII
Buckmark
Both are fun handguns to shoot, and great to learn on. They both cost a fairly similar amount, magazines cost about the same, are capable of being very accurate once you learn to use it, and both function fairly reliably with only 4-6 misfires or misfeeds per hundred rounds of cheap ammunition, and 1-3 with high velocity ammo like yellow jacket or otherwise.
The one problem with both is that they are a real pain in the butt to clean, with the Ruger being the worse of the two. With the MKIII, it is a very tight, difficult pistol to clean, and even though I have done it lots of times, I still find myself taking around 30 minutes to do it, whereas I can usually do my other firearms in 5-10 minutes.
The buckmark is far easier to clean, but compared to most any centerfire pistol, it is still a pain in the butt, as you need an allan key handy just to get the slide out.
Magazines for both are a little bit of a pain as well. They have a little thumb tab on the side to push the follower down to load the rounds, which sounds great, but after loading that magazine 5 or 10 times, your thumb will start getting really sore. That tab is stiff and unergonomic.
So those were the two I started with. Both had pros and cons, but they did the job and taught me the basics of shooting.
Now that I had learned the basics of handguns, I picked up a centerfire pistol with the conversion kit. I purchased a Kimber 1911 chambered in .45, with a Kimber Rimfire conversion, both of which are nice.
Depending on what type of handguns you like, there are 1911 Conversions, Glock conversions, and I am sure there are others out there as well. I happen to be an all out 1911 guy, since I like an all metal handgun, but there is plenty of choice out there.
The fit and finish on the conversion is super, functions just as reliably with both high and standard velocity as my buckmark and MKIII, magazines are the same cost, and accuracy is just as good, with groups in around 2" at 10 meters. You get to train with the same trigger you use on your bigger calibers, which
trust me, is hugely benificial. It takes 45 seconds to swap the slide and swap the magazines, and you are good to go.
Cleaning is far easier then most dedicated pistols. Just watch and video on how to field strip a 1911, and it is exactly the same procedure. No tools needed.
Magazines are far superior in terms of usability. They function just like a standard single stack magazine, so no thumb button, but it loads very easily from the top. It is actually easier to load the rimfire magazines to the 1911 conversion then it is to load the .45 magazines for the same gun.
Having done both paths, my recommendation would be to go for the conversion kit. It will come in right around your budget, and it is the same to shoot as your larger caliber, easier to load, and
way easier to clean.
That being said, you still can not go too wrong with a dedicated .22lr pistol, and you should have one in the collection, but I think you would be well off with a conversion kit and a centerfire to get you training with both .22 and a centerfire on your budget.
Let us know what you choose, and happy shooting.
Cheers.