Exploring the area outside of Luang Prabang is as easy as pie (and, very cheap). Borrow a map, rent a motorbike and whala you're all set.
The surrounding countryside is lovely and all of the sites are well worth a visit, but there is one that clearly stands out from the rest - Tat Kuang Si (Kuang Si Waterfall). This multi-tiered waterfall is truly stunning. Sparkling blue water rolls down an endless number of cascades with the largest being a 25 meter plunge. The water softly flows at the lower part of the falls and powerfully rushes at the top. Swimming opportunities abound in the various pools along the way (including Tarzan like rope swings for those in need of an adrenaline kick).
One word of advice though - the hike up to the top of the falls has a very steep section where it's easy to lose your footing. Make sure to bring proper sneakers, or risk a nasty fall.