I would suggest starting with a .22 rimfire - range time not dependent on buying somewhat spendy centerfire or reloading centerfire. You'll do a lot more shooting if a .22 is in the quiver; you can trust me on that.
So I would start with a .22 while getting some good solid shooting basics down. After that, you'll also have a better idea of what appeals to you and what doesn't.
Aside from the usual suspects in the accurate .22 rimfire category, there are a lot of new .22's out over the last five years or so, from various manufacturers. For example:
https://www.gunsamerica.com/blog/brownings-miniature-rimfire-1911-22s-new-gun-review/
http://www.gunsumerreports.com/review_smith_wesson_M&P22_p1.php
So there you have a single and a double action example. I like single action, but that's for everyone to decide on their own, without somebody telling them what they SHOULD like.
And finally, you can choose a platform that offers a conversion kit in .22. I have an extremely accurate Hi Standard Supermatic Citation that I've owned for nearly 40 years, but if I was doing it again and knowing what I know know, I would have gone the conversion route instead. I have a .22 conversion for the Browning High Power, which works with both my 9mm and 40 S&W variants, and for my Dan Wesson 10mm CBOB I have a 40 S&W barrel and spring, as well as a .22 conversion kit. This doesn't work on the revolver side of things of course, where the closest thing is to have something like S&W K frames in both .22 and whatever centerfire.
The catch with going the conversion route is you have to have already known what platform is the one that calls to you, AND ensure that platform has .22 conversion kits available, AND you are going to have a much greater initial investment.
Except for conversion kits like the Marvels for the 1911 platform, it is pretty unlikely that a .22 conversion will be a tackdriver like a full on target .22 - the Marvel kits are crazy accurate, but the price tag reflects that. On the other hand, a .22 conversion kit allows you to have the same grip, the same trigger, the same ergonomics, etc whether you are shooting the centerfire caliber or the .22 rimfire. And you can change back and forth at the range in just a few seconds. There's a lot of good things to be said about that.
But either way, I would recommend starting out with a .22 rimfire in the quiver and expanding from there.