Does transferring a domain name automatically transfer all DNS records? I'm transferring to GoDaddy. I'm particularly concerned with the MX records and others which one creates for Google Apps Email and other services.
Generally, no. It's important to understand that the domain only references a nameserver; the DNS records aren't actually stored with the domain registration at all.
When you transfer a domain name, most gaining registrars will leave the nameserver associated with the domain unchanged, so your DNS records will keep working as long as your DNS does. If you are using an external DNS provider, you don't need to do anything else. But if your previous registrar also provided your DNS, now is a good time to manually copy your DNS records somewhere else.
The reason that a registrar can't automatically transfer your DNS records is because most DNS providers have zone transfers disabled. Without zone transfers, there's no way to know all of the subdomains you might have configured.