1. Get your "zonefile" from your current name server provider.  A zonefile is just a text listing of all records for a domain.  It will probably include entries for the name servers, you will never set those though.
  2. Make sure you can login to your Network Solutions account.
  3. Change the domain's name servers to Network Solutions
  4. Add in the records (A records,  CNAME records, MX records, and any others) from your zonefile.

Network Solutions articles:

Move DNS To A New Server

  1. Within Account Manager, select My Domain Names
  2. Select the domain name you want to change and select Manage 
  3. In the green box, select Change Where Domain Points
  4. Select Domain Name Server (DNS)
  5. Under Specify Domain Name Servers section, type in Name Server 1 and Name Server 2.  To enter additional Domain Name Servers, click on the Add More Name Servers link.  To use all Network Solutions® name servers, click on See More icon to the right of the name server boxes and select Network Solutions Name Servers. 
  6. Select Continue, then Apply Changes to accept changes
Note: Domain Name Server changes may take 24 - 36 hours to update throughout the Internet.

Edit Domain Name Servers (Changing the domain's name servers to Network Solutions)

To use all Network Solutions name servers, there is no need to enter any name servers, just click on the Move All to Network Solutions button.

Note: Name server changes may take up to 48 hours to update throughout the Internet.

Extra Note: Yes one says 48 and one says 24-36, often we see much faster changes.

How to Manage Advanced DNS Records (Add in the records)

This talks about A records, MX records, CNAME records, TXT Records (SPF too), and SRV Records.