Welcome to Graybeardoutdoors. Stick around and enjoy the site.
Well as far as your selection, as Mac said you didn't mention the actual "bridge" you plan on making. Be it Hunting or Target.
Also you didn't mention if you handload or not

This would be a
big factor in my reccomendation.
And If not..1st off why not?

... then, are you looking for a inexpensive cartridge to shoot?
The 223 is always a good bet in my book, as I own 2 and love them both. Awesome shooters, and easy to reload for. Easy to stretch out a Ib of powder and componets are widely availible. If you don't handload, then ammo is everywhere and cheap too. Just a good over all inexpensive performer, handload or not.
if you want to shoot often, then I wouldn't recomend a 204 or 22-250
unless you handload for it. That or if you have lots of money. In that case go for it

But these rounds usually are more expensive than 223s. Same kinda goes for the 243 and 25-06.
Give us a little more info on what you are actually looking for.
-Aaron