RMNP is very crowded that time of year, but if you work a bit you can stay away from crowds. There are multiple entrances and the one with 80% of the traffic is the eastern one near Estes Park.
Only 14er in the immediate area is Longs, which is definitely hero stuff if you've recently arrived from sea level. I've been here almost 10 years and I'd have some trepidation about the length and exposure/fall potential near the top. Having said that, the 13ers in the area are gorgeous and MUCH less popular. And views are spectacular no matter where you go. Hike to anything above treeline and look around (treeline is 11400 on average). Sierra's point about Gerry Roach's books is spot-on, as is the website (there is a website called 13ers.com too, with the same format as 14ers.com).
I have a friend who's done all the 14ers at least once. His rule for flatlanders is to sleep at least one night above 9K before even trying a 13er/14er. My major advice is to get an insanely early start, go 70% of the speed you do in New England, and if you think you're going too slow, slow down. With enough daylight and patience you will make it. All the caveats about weather apply, of course. Water is generally unavailable above treeline, so bring your big Camelbak. I drink 4L over a full day hike and still need to catch up afterward.
Hoping for a less hot and wetter summer this year. Winter has been good to us, hopefully the next 6 months will be too. Enjoy!