By far the easiest way to get to Luang Prabang is by air. Thailand's Bangkok Airways flys almost daily from Bangkok to Luang Prabang. For domestic air travel, Lao Aviation is the only carrier, and flies to Luang Prabang from both Vientiane, Phonsavan and other places. Lao Aviation had a somewhat spotty safety record in the past, but has taken steps to improve the situation and upgrade its fleet. See my review of Lao Airlines as well the guide to Luang Prabang Airport.
Luang Prabang Airport
Road travel used to be only for the most adventurous, as few highways were paved for their full distance. The road between Vientiane and Luang Prabang is now completely paved, and there is regular bus service between the two cities, as well as Vang Vieng in between. Note however that even with the roads paved, the trip from Vientiane to Luang Prabang or vice-versa is nine hours.
Boats up and down the Mekong are an increasingly popular mode of travel. You can travel downriver from Thailand to Luang Prabang as well as upriver from Vientiane. Although speedboats can make the trip from the Thai border in just a few hours, this is probably a trip best taken slowly. Various operators offer two and three day cruises between Chiang Kong in Thailand and Luang Prabang.