You don't need transport in Luang Prabang. It is such a small compact place that you are never really far from where you want to go.
If you plan your day effectively you won't have to walk long distances between your accommodation and anywhere.
If you are hot or tired stop at a cafe or restaurant for a rest and a drink before going on your way.
Walk slowly, nobody rushes here.
Coming from the south (vientian and vang vieng basically) the cheapest way to get to Luang prabang is by bus. I heard it's definitely not comfortable: buses are overcrowded, slow, and the road is twisting and turning - when it's not washed away. Having heard many horror stories, through a travel agent I managed to organize a group and rent a mini-van - making the trip a lot more bearable. Still we had our rough patrches, like trying to navigate a river of mud while avoiding to fall down the ravine where the road had just vanished.
If you insist on taking the bus - buy your ticket in advance: buses are often full and there are only a handful of them a day
Coming in from the North the logical way of getting to Luang Prabang is by boat - slow boat if you please. it takes a total of 18 hours not including an overnight stop in Pakbeng but it is so much safer (and scenic) than fast boats. The entire boat portion cost me 550 bhat (about 15 $). Slow boats officially carry 25 passengers + cargo (rice, normally) but we were about 50 people (+rice). They are very safe, which is a plus since there are no life-vests on board and the mekong is everything but peaceful. The noise of the engine can be a bit annoying, but you can go and sit outside on the deck: it's quieter and you can sunbathe.
(pic left : bicycle in Luang Prabang)
I did an overland travel from Bangkok to Vientaine by bus. It was a great journey, basically slept throughout the way from Bangkok till the next morning in Nong Khai. From Nong Khai there are tuk tuk service for about 20-30bt to the Thai Immigration Complex. After getting your passort stamp you can easily hop into a mini bus that service both the Thai and Laos immigration complex across the Friendship Bridge.
On the Laos site there is suppossingly a bus that run from the the complex to Vientaine but I took a taxi for 60bt per person.
I would not say this is the best but I wanted to see the Friendship Bridge.
Around Vientiane, I would suggest you to rent a bicycle as the city isn't very big but its really dusty. But you will be able to cover the whole Vientiane in a day with a Bicycle.
In Luang Prabang, a bicycle will also do the job but if you are going to Pak Ou Cave (don't waste your time and money if you are travelling alone, too far and expensive to be on the boat) or Koung Si Waterfall you might be able to experience the boat on the Mekong River. I was on a shared boat and paid USD7 per person for 2 days to Pak Ou and Kong Si Waterfall.
Tuk Tuk is also available around Luang but why waste the money if you can walk around the town by foot.
Overall, boats were the best mode of transportation. Make sure you have enough time on your trip to use them.
The last leg of our trip was by slow boat across the Mekong to the Thai border control checkpoint. We were happy and relieved that we had made it. Laos was a great adventure!
Here we are getting ready for the first leg of the journey. We tried not to think of the people who were killed when their boats hit rocks in a shallow part of the river the previous dry season.