The license you obtain is good for 90 days and you can be married anywhere within the State. You provide the Officiant with the license and they will sign it and file it with the Town Clerk who issued it. Approximately 15 days later, your Marriage Certificate will be available from that same office, the fee is currently $12.00, I believe. You will need this certificate as it is the legal document declaring your marriage and is a statistic of vital record. Your next of kin will have changed, and it has far reaching effects on financial arrangements. You should make copies of it to send to all relevant bodies to support your name change where applicable. It is your responsibility to do this. Click the link for a handy name change kit. [This is a third party service which involves a fee]
I will file your paperwork by certified mail to the relevant office and I also keep a copy for my records, a copy is sent to the couple for their records at the same time. This has no legal relevance for you, but I feel it completes your keepsakes of the day.