What usually happens to government bonds after they are sold by a city, state or county to an investor? 

They are re-sold on the "secondary bond market" depending on how much of a risk they are as determined by bond rating agencies. 

They are held by the buyer who collects interest and eventually is repaid 

They are purchased by banks, insurance companies and pension funds to provide income for their own customers or clients such as pension funds. 

All of the above can happen