What type of solder should be used to assemble Adafruit cobbler
Anything really if it is a home project.  60/40 Tin Lead is really easy to work with and is your off-the-shelf choice.  This only becomes an issue if you plan on selling your Cobbler.  In this case you may need to go to lead-free solder depending on your country rules and regulations.

If this is your first time, pick up some solder wick (you'll need it) and either buy solder with a flux-core or buy some flux separately.