Few municipalities and counties do a lot of bespoke software work -- most of their contracting hours go either to configuring and integrating COTS systems that handle specific aspects of their operations, or doing reporting out of those systems. The lion's share of their IT dollars go to licensing the systems themselves, so if you really
want to get into the municipal space, find an underserved departmental need and establish yourself as a player in that space.
States are more complex, as the agencies are larger and generally have more specific needs; you can do more custom software work, but your best bet is to get embedded in an agency and learn their processes and systems extensively. I don't recommend trying to break out of a governmental "level" (local to state, state to fed) -- totally different contracting environments, and your time is best spent learning your given market very well.
After that, it's a lot of sales efforts and searching for/responding to RFPs. Your best bet is to get onto state purchasing lists (similar to GSA, but generally with a lot less red tape), as most states will let cities and agencies skip the bid process if they're buying off an approved vendor list. If you have to go through bid, the very best thing you can do is get a champion with purchasing authority and let them know exactly what your product does that the competitors don't -- it's very common to write bids so that they end up targeting a single vendor, but you have to have valid and meaningful differentiators (not always technical -- don't forget that bids encompass not only software but also support and consulting!) for the bids to pass muster. If you're looking for competitive intelligence, you can generally get access to city and state contracts under sunshine laws -- these will tell you not only what the pricing is in your market, but also what cities and agencies are looking for in their bids.
This may or may not apply, but female-, minority- and veteran-owned businesses (so-called "historically underutilized businesses," or HUBs) are usually a plus when bidding, as are small disadvantaged businesses. I strongly recommend checking out getting SDB or HUBZone certification from the US Small Business Administration if you think you qualify -- it'll open some doors for you.