A great way to spend a day in Hue is to rent a motor bike, head to Thuan An beach, and when you're done, just keep going south.
You can see on a map that Thuan An is an island at the top of a long, slender string of land, with the China Sea on one side, and the waterway feeding into the Cau Hai Lagoon on the other. It's an easy trip if you don't know your way around because there's only one road so it's impossible to get lost.
On the outskirts of Thuan An you'll see an endless array of Vietnamese-style mausoleums, and as you continue on down the road, the local character of the area really starts to come out. The road seems to be clogged with students on bicycles almost no matter what time of day or day of the week, and you'll get plenty of 'Hellos' as you drive by -- foreigners are still pretty rare in these parts.
The road leads to Vinh Hien, where there is a newly-constructed bridge crossing the lagoon. You used to have to take a ferry, which was lots of fun, but those days are gone.
Once you're across the lagoon, there's a rough, winding road alongside it through the hills, with stunning views of the water along the way. It eventually crosses the railroad tracks and hooks up with highway 1A. At that point you can take a left if you like and head down to Elephant Springs, or a right to return to Hue -- you'll find Bach Ma National Park on your left along the way, but that's a day-trip in itself.
The loop around the lagoon is about 100 kilometres all told -- expect it to take at least two and a half hours to finish, but then, of course, what's the rush? It makes a pretty great trek for experienced cyclists, too, though that's a full day -- you can always overnight at the guest house at the base of Bach Ma National Park if you run out of daylight or steam.