Do you know about the Neighbor Joining method? It's great in that you can actually do it by hand, and my professor found a tutorial online that allows you to walk through it by hand.
http://www.icp.be/~opperd/private/neighbor.html is a really good tutorial, and I believe it uses the Jukes-Cantor substitution model if I remember right. It doesn't teach you about testing the node, like bootstrapping--one of the most basic. You could do bootstrapping by hand, but I wouldn't recommend it considering you'd probably do at least 500 bootstrap replicates per node. Just know how to do one, then use
MEGA do all of the calculations for you.
The best way to learn phylogenetics is by just doing. Download MEGA and
MrBayes, do the tutorials, and get the trees. I'm sure you've heard of
GenBank by now. Develop your own miniproject by grabbing sequences off of GenBank and use the various tree building methods in MEGA and use MrBayes to build trees. See where they're similar and we're they're the same, look at the values of the nodes, when should you more comfortable with the nodes? Does a high value mean a correct node? In distance trees, what is long branch attraction?
Know what Maximum Likelihood, Bayesian Inference, and Bootstrapping are supposed to do, but not the math behind it. Know what Markov Chain Monte Carlo does, but not the math behind it. Know the various substitution models, and know the math on the ones that aren't so parameter heavy.
Most importantly, know that if you see a tree in the literature, realize that it may be wrong only because of poor alignment, and it may be something that is out of their control, and it may be an issue that can be solved by knowing a little more about the organisms and how they got to where they are and such.
Hope that helps.