Starting at the state or local level, or in the private sector, can be helpful, but many contractors have started directly in federal contracting, including, most recently, Nava and Ad Hoc.
If you want to get into federal contracting, you need to start networking and researching how a particular part of the federal government operates. Pick a particular department, and start there. The biggest spenders are the DoD, the VA, and CMS, so most people start out focusing on one of those agencies, although there may be some advantage of targeting a smaller agency where there will be less competition. It's hard to overstate how much of a vendor community there is around each of these agencies, but you can start following industry newsletters and events that are specific to any one of the above agencies, although most of them will be in the DC area.
The other thing to note is that almost everyone starts off in federal contracting as a subcontractor. It's fairly unusual for a company to land its first federal contract as a prime, and this would require substantial networking in DC. Your best bet is to start networking with contracting firms, and see if you can find a niche. Once you're inside an agency, you'll have a better sense of what your next steps should be, and how you would take the leap into bidding as a prime.
Finally, it's worth noting that the federal government has multiple small business programs that you can take advantage of. The most valuable and hardest to qualify for is the 8(a) program, but there's many others for veterans, minorities, and women. Also, no matter who your founders are, you can always qualify for the HUBZone program. However, to qualify for any of these programs, you're going to need to plan for that from the founding of your business.
Anyway, I hope that helps, let me know if you have any more specific questions.