I've never done it on a bike, but in a cage a couple of times...once on a backpacking trip where we took the train out of Durango, and once on a photography journey. Either direction is good. Places to stay...Durango is a small college town, so there are some cool bars and places to eat. Ouray is on the other end of the million dollar hwy, coming in from Montrose. Ouray is called "Little Switzerland"...small town surrounded by 13 K peake, hot springs, and a cool old hotel that has a Mexican Restaurant and nice, funky bar. Neat little downtown area...population is about 1K for permanent residents. Durango probably has about 10K residents, when college is in session. Durango is also closer to Mesa Verde, NP, as JC mentioned. Very much worth seeing if you have the time. The old mining town of Silverton is at the "top" of the Million Dollar Hwy...there's a neat old hotel there, a couple of cool bars, but it's HIGH...I think the elevation is between 10-11K in town. You can do a nice loop if you base out of Durango, go to Mesa Verde, then up through Cortez (I think that's the name), to Telluride (really cool place), through Dallas Divide, then to Ouray, on to Silverton, then back down to Durango. There's a NP lodge in Mesa Verde, if you wanted to spend a night there. From the Ouray side, you can head to Gunnison, Crested Butte, through the mountains over towards Aspen, see the Maroon Bells (big mountains). Take a good camera...there is MUCH to photograph!! Watch for fallen rock between Silverton and Ouray...you can't really go fast anyway, but do watch the blind curves. There are 1,000 foot drops, straight down, just a few feet from the road, so screwing up would be a big one. The whole area is spectacular, with 14K peaks all over the place. Stop in Molas Pass on the Million Dollar Hwy if the weather is good. Be prepared for thunderstorms in the afternoon, especially up in the mountains, and if it's lightening, take cover, cause you're above tree line. If you like taking photos of old architecture, spend a little time in Silverton...lots of old buildings, old Bordellos, etc. It's not a big town, so it doesn't take long to see everything. Good eats and beer though
You will be traveling through the San Juan Mountains, Winmenuche Wilderness...all spectacular. Here's a few photos of what you can expect to see: