Your state's court website should have the basic info that you need. As a general rule, if the birth father is listed on the birth certificate you do need to notify him and get him to sign the name change form. His signature needs to be notarized. If you can't find him, you have to prove that you can't find him and there are specific steps to follow to prove that he can't be located. After you follow those steps, then you can publish a notice in the newspaper of his last known address and then after that runs, that is considered notice and if he still doesn't reply, you can proceed with the name change.
If the father isn't on the birth certificate and hasn't been named as the father in any court proceedings (child support, visitation, paternity) then he has no legal rights and you are free to proceed without his permission.
In either case, you'll need certified copies of the birth certificate so order one if you don't have one already.
ETA: Please ignore the response that indicate that your fiance needs to adopt your son or that your son's birth father has to give up his parental rights for you to change his name. Those statements are simply not true. All you need is his notarized permission for the name change on the name change petition. This permission has NOTHING to do with parental rights or adoptive status. I know this because my husband initially agreed to a name change for his daughter when she was 5 from her mother's maiden name to her step-father's name. When her mom left the step-father, we then changed her name (with the mother's agreement) to my husband's last name and that will be her last name until she's an adult. Neither of these changes were related to changes in custody, marital status or adoption.
For your younger son, you and your fiance can fill out the forms to change his name to whatever you want, whenever you want. You can do that before or after your wedding or have both boys done at the same time once you get the older boy's father's permission (if he is legally recognized as the father - if he's not, then you can just do both boys together).