The world is a very big place, and most of it is probably not the country you live in, so you may want to be a bit more specific.
In the UK, different govt. departments and other public bodies will have different requirements so there isn't an over-riding list of things you can do to be eligable for all contracts.
In general terms, you may have to be part of several bodies in order to bid for work, for example, I work in the college sector and to bid for a serious security contract I was looking at, you'd probably need to be registered as a supplier with the crescent purchasing consortium (http://www.cpc.salford.ac.uk/) and be able to bid on contracts published via the Official Journal of the European Union (OJEU).
The mechanism for other contracts to other groups within the UK public sector will be similar to at least some degree, but the devil is very much in the details.
As for standards you have to meet internally, these again will vary depending on who and what. We're likely to be as interested in your financial health as we are in exactly how many bits are in your database's encryption key.