As already mentioned, there's Big Thompson outside the park. There are dozens of opportunities inside the park, and I haven't found those waters to be too crowded -- even the ones close to the roads and entrances. Of course, if there's a parking lot and a well-worn path heading to the stream, you're probably not the first. As a general rule, the farther away from Estes you go, the lighter the traffic; and the west side of the divide will be less active than the east.
There are a lot of different types of opportunities too. Meadow fishing for large, wary trout (See avatar - that's the Thompson inside the Park), pocket water and/or high gradient fishing for opportunistic smaller ones -- all fairly close to the road. There's also an opportunity (though slight) of being mauled by a bear, attacked by a mountain lion or 'gored' by a moose. But seriously, it's cool just to see any or all of them. On the west side, there's the opportunity to fish the headwaters of the Colorado, and with a moderate hike, get to a place where you can jump across.
If you're into numbers, there are places where it's relatively easy to hook a brook trout on every cast. If you'd like to add one or two species of native cutthroats to your life list, there are Greenbacks on the east side and Colorado River cutts on the west side of the divide. You'd have to do a little homework to find them, especially the latter. Most streams have either rainbows or browns. Grant at Estes Angler or Scott's Fly Shop will help you narrow things down a bit.