The easiest ammo to find would be 22lr, 9mm, 223 and 12ga. 308:to a lesser degree. Plan accordingly.
Yes 22lr is not adequate but is still deadly at close ranges and better than nothing.
If you live where AR15s are banned you might be able to find a Ruger Mini14. They are in 223 and also Ruger makes in 7.62x39. Not generally quite as accurate as an AR, they are extremely reliable and easier to clear a jam due to the Garand style action. I have both a mini and an AR. The mini is a bit more handy. Reloading is a tad more difficult.
.22LR is great for practice practice practice. Yes: if you're going to have only three guns, and one of them is a .22LR, there will be compromises elsewhere. Doesn't matter: the practice with the .22 will pay of when TSHTF.