hey. i've just came back from bolivia a couple of months ago. i stayed there working for 3 months and went travelling for one more month with my boyfriend. i got to know bolivia and the bolivianos quite well, so i guess i can help you out a little bit.
first of all, travelling indipendently is not a problem at all. most of the places are reachable by flota (in bolivia used word for bus), whenever it takes ages and a lot of patience to get to some places off the beaten track. infrastructure is not quite developed yet. there still exist some dirt roads where a bus needs 24 hours to get to the place instead of maybe only 18 if it was a proper road. but there are as well places like the famous salar the uyuni, where you have to take a tour (min. 4days, 3 nights), to get around to see all the things. or if you wanna go to get to know the amazonas, you need to take a tour as well.
talking about means of transport. you can fly to almost every bigger place in bolivia. but some of these places like rurrenabaque in the amazonas only have a dirt-airport which means that planes cant land in the rainy season. as i mentioned before, public transport is quite good in the sense that you can get everywhere you need. its cheap as well. there are also posibilities to buy tickets for tourist buses in the hostels you might stay. there way more comfortable and sometimes faster than the flotas used by the bolivianos. but i think its worth travelling by public transport the bolivianos use, so that you can somehow also live their life and not be the 0815-package-gringo.
if you wanna do stuff indipendently: learn some spanish!!! otherwise your a welcome victim in the sense of getting ripped off whenever they can (taxi, markets, buses etc.).
ok, now to your interests and what you wanna see there. as you know, bolivia has everything but beaches... and heaps of outdoor activities. if you like biking, do the deathrail by bike going down to coroico. but watch out that you get a reliable company with safe bikes! in coroico you can go canoeing, as theres many rivers. in the amazonas as well. talking about the amazonas. its amazing to go hiking there. rurrenabaque is one of the most popular getaways to discover the rainforest and there are a lot of different agencies which offer different tours. i would strongly recommend you CHALALAN! its an indian community who developed an ecotourismplace in their community together with an NGO from holland (?). from my opinion (used to work in la paz with tourism) it is the only place where they really act eco-friendly. every agency says they do it, but none really does. its a bit more expensive though, but amazing.
You wrote that you will stay in LP for a couple of days. I strongly recommend you to stay in the cute little hotel called El Consulado. its about 50$ per night but a unique diamond in LP. But of course, this depends on how much you think to spend. There are of course cheaper places at the tourist road Sagarnaga.
Well, i already wrote so much and maybe a bit unorganized. but i hope it helps you a bit. i have a lot more to tell, but i guess space is limited. so if you have a concrete question, dont hesitate to ask me.