I need to transfer a few domains to godaddy.com. All of the domains are with networksolution, but emails are sent to one mail server. When transferring to godaddy, and I choose to use godaddy's NS, would this affect the mails being sent if I create MX at godaddy pointing to the current mail server?
It doesn't matter where your DNS is hosted, provided of course that you have your nameservers correctly set for the domain. The MX record(s) can point to anywhere you like.
You may experience some downtime during the transfer as GoDaddy picks up responsibility for the domain. Are you using Network Solutions' name servers currently? When the transfer occurs, Network Solutions may remove the DNS mapping and depending on DNS caches on the Internet it may take some time before requests go to GoDaddy for resolution.
Mail delivery should be okay because the sending side should retry on DNS resolution failure.