I just got back today from spending about 5 days total on the Lofoten Islands. It is indeed a beautiful place, even in the late winter/very early spring.
As far as what I did, I spend most of my time on vestvagoy, flakstadoya, and moskenes islands. They were beautiful, especially vestvagoy and moskenes. Flakstadoya would've been spectacular, but for much of the time it was just clouds for me there.
For getting around, at least in the winter schedule, the buses are definitely not the most convenient. They only run 1-3 times a day to most major places. Thus, if you are rather short on time and want to see the most, it's not the best. Still impressive though, that they have the public transport system in place.
So, I decided to rent a car. The rent a car that I had wasn't the cheapest, but it made it convenient. I could've gotten a rent a car a lot cheaper had I known how to drive a manual instead of automatic. There aren't many automatic rent a cars on the island I guess.
By the way, if you happen to stay in Stamsund, there is a used car rent a car that will rent cars out for 450NOK a day, or 350NOK a day if you rent for more than 2 days. That was the cheapest I found on the islands.
As for lodging, I stayed in Hostels. Namely, I stayed in the one in Stamsund and the one in A. They were both really nice places to stay at. The one in Stamsund is one of the cheapest hostels in Norway I think. Still not "cheap" though if you are thinking of anything near hostel prices in Asia or South America. Had it been warmer and not the end of the winter season I would've looked at camping as that is indeed even cheaper and you pretty much stay where you want, but I'm glad I hadn't as a couple of the days had gale force winds and downpours.
Anyway, all this to say, you can do the Lofoten islands on a budget, but it might be rather difficult. If you can rent a car though, do it, as you see much more and get to stop when you want. If you are taking the bus, you'll be trying to take photos out the window which doesn't lend itself to good photos... :)
Hope this helped, and if you have any other questions, let me know.