That all being said, going out and playing with some of the established tools, and reimplementing some classic models (or building models off of wet lab papers, or whatever) is going to build you up a great skill set, and make it a lot easier to find a lab position if you want to go that directon (either a paid one or a volunteer one, each has advantages).
I'm personally enough of a biologist to feel compelled to point out that a lot of what has been done in larger networks has diverged from biology in critical ways - some of it might be interesting in its own right, but it's not really neuroscience in any meaningful way.
Get a solid grounding in Neuroscience. (Kandell, Jessel and Schwartz, Principles of Neuronal Science, is the standard text, it's excellent, and highly torrented.) Please, please, please take some time to understand the variety and complexity of single neurons - they are way more complicated than many of the people who model systems with high numbers of neurons let on. Having a system with 100 billion simulated neurons means an awful lot less if the neurons themselves are shit.
Re-implement some classic systems from scratch. Yeah, I mean start with Hodgkin and Huxley, and build up from there. You will learn things from doing that yourself that you'll miss by just diving in with established tools. (And a lot of the established tools have issues.) Itzikevitch, Wilson, and Trautenburg are all favorites of mine off the top of my head. Strogatz is great as an introduction to dynamical systems.