If you enjoy getting up close and personal with nature than you definitely want to drive the Old Fall River Road. The road is narrow and slightly curvy, but be assured it is safe.
At the RMNP visitor centers they say this road is a long way through the park but it is worth it if you have time. The entire road seems to be uphill and would be fine for most vehicles but be careful about overheating! Many scenic overlooks can be found along the drive and they are worth stopping at. Not a "safe" road by any means, but just be alert and pay attention to the road.
This is a steep, one-way (up) mountain road that goes from the end of Horseshoe Park up Fall River Pass, and meets the Trail Ridge Road at the Alpine Visitor Center. The road is curvy, narrow in places, and without amenities like guard rails and curbs, so you have to pay attention to it. Which is too bad, because it's a beautifully scenic drive. There are several overlooks back down the valley, and the Fall River runs either right alongside the road or within earshot, with a few pretty nearby waterfalls that deserve a look.
It's a fairly smooth road, but relentlessly uphill. Any SUV can make it, as well as any front-wheel drive family car. There are no restrooms until you get to the visitor center at the top, so be prepared. The road is only about 10 miles or so long, but will seem a lot longer, and there are plenty of pullouts so that those in a hurry can get by those who aren't. (The speed limit is 15 mph.) The road is closed to vehicles in winter, which is roughly October to May or June, at which time it makes a nice hiking trail.