I want to route my.domain.com to the url of my lambda function (which I setup and working via the API Gateway). I can't seem to be able to do this. I also can't seem to be able to to CNAME to anything (so my.domain.com results in a dns error when I CNAME it to domain.com). The normal url (domain.com) is routed to a static S3 bucket and is fine.
The domain was bought on GoDaddy. I changed it so that my GoDaddy Name Servers point to those in a Host Zone which I set up via Route53.
You can't make a CNAME on domain root ie. domain.com because it's zone apex.
To be able to do what you're looking for you need to make a CNAME example.domain.com pointing to CloudFront endpoint generated by API Gateway.
You can follow more detailed API Gateway domain setup instructions.