If there is no father listed on my child's birth certificate, how would my fiancé go about adopting my child?

In most places you will have to be married for no less than a year. At that time a reasonable search of the biological father will have to be made in order for him to terminate all parental rights.

In most places (but it can vary) not having a father listed, can make it more complicated, and it will be more costly. By the rules of the court, you will need to attempt to locate the father, prove he is father normally by DNA test, get him to sign the rights over for adoption, or file against him in court.

The process will need an attorney.

As noted, not having a father listed will make this harder. Also as noted, you will probably need to be married for a period before he can enter a petition for adoption.

So I would recommend you start shopping around for a Family Law attorney now. A local attorney can advise you what requirements you need to meet and what you will have to do to clear the way for your husband to adopt. You will also be able to get a cost estimate so you can set money aside.