You will outgrow 300cc in very short order if you are an actual adult looking at street bikes. I'd start at 500cc minimum, particularly if you are north of about 170 lbs.
Take the MSRF course. It will get you an insurance discount, give you credit for some or all of the test (don't know if CO requires both written and riding exam. MSRF course counted for both in FL), and will teach you some really useful skills.
Best of all, you can wreck somebody else's bike instead of 'learning' on your own.
Buy and wear good protective equipment. Not something to cheap out on.
I ride my 500cc bike up to Estes Park and back whenever the weather is nice. Fun to ride the curves in Big Thompson Canyon. I wouldn't put it on I-25 for anything, as it tops out at about 80 MPH and takes it's time to get there. More modern bikes in general, and sport bikes in particular, can do more with less displacement, but 500cc is still as small as I'd ever go. Personal opinion only, but 800cc give or take is a really good displacement for a general purpose bike you aren't going to ride very often or in the same fashion as a stupid 20 year old.