.223 is a good 2nd rifle caliber. 308 is a good first rifle caliber. Become competent with a 308 and as you gain experience and become reliable with your shots you will find that having a .223 as well will become attractive. Its a great "as well as" caliber, but not universally as useful like the 308.
I would buy new, but if your budget doesn't stretch that far there are often good Tikka 308's on the market at a reasonable saving. Don't skimp on a 'scope to buy a new rifle. 20" is a good barrel length for a 308 to add a suppressor to. I wouldn't go shorter - you are giving performance away for no good reason.
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