While I sympathize with your plight and Chris' recommendations are rock solid, I'd say that a couple years working helpdesk doesn't automatically qualify for promotion.
The structure of your IT department and your general office politics have the ability to greatly influence the focus of my recommendations.
What I am looking for here is what can I do to better myself to inside/outside places to move up this ladder?
A proven track record of higher level skill, a clearly communicated desire to move outside of current role without resentment, and growth outside of your existing skill-set is an absolute necessity from my perspective.
I take what I can from my job - working on projects that I can learn from - but what projects should I be volunteering for?
What are the responsibilities of the people who have the position you want? What don't they want to do that still falls under the role you want? Do it for them. Do anything you can for them. What projects? Any and all that you can. Having the existing IT staff purporting your viability for the position will absolutely help you with your career growth.
What can I do for myself outside to make me a more viable candidate for a systems admin?
This is going to be unique to your environment. What was the job description they were hiring for? If it was mid-senior level, you're simply not going to be up for serious consideration without a very good reason for it. If they are looking for entry level, what were the areas where you were not qualified?