This may not be necessary depending on the type of developer you choose. For example, if you are contracting with a professional website development firm, it could manage the entire process for you. It would be good to understand how to Register the Domain Name in case you change companies at a later time. However, you may choose to register your own domain name so that the city maintains control rather than giving that control over to someone else.
The registrar is the retail company from whom you purchase the rights to your domain name. This company should be accredited by the Internet Corporation for Assigned Names and Numbers (ICANN) at http://www.icann.org or by a national ccTLD authority to register Internet domain names. ICANN has authority over all the gTLDs, while the Internet Assigned Numbers Authority (IANA) at http://www.iana.org has authority over ccTLDs.
The registrars for the gTLDs are numerous and the pricing is very competitive. Here is a link to the list of ICANN accredited registrars: http://www.icann.org/registrar-reports/accredited-list.html. Some that you may recognize are GoDaddy and NetworkSolutions.
For your DotGov registration you can start here https://home.dotgov.gov. The following page https://home.dotgov.gov/registration/requirements/ will show you the list of new requirements that apply to new .gov registrations as of April 26, 2021. The cost of registration fees are being waived through FY2021 which makes this a great option for using as your domain.