I don't think that NSLOOKUP or Dig can provide your local DNS servers, as this is an OS call. This is (usually) most easily done by
on unix systems, however it looks like doing an nslookup will tell you the nameserver used for the query, so
will return a "Server:" line which has the server IP which was queried.
To query DNS servers for a site using your local nameservers -
dig sitename.com ns
You can do the same thing using other nameservers (as long as those nameservers will allow you to do the lookup) by using @domain name - for example dig @220.127.116.11 facebook.com NS will tell you Facebooks nameservers according to the nearest Google Nameserver. You could also do dig com NS to find a list on nameservers for .com domain names, then query one of those nameservers to see what the authorative nameservers think.
To query a record type
dig sitename.com RECORDtype
For example dig sitename.com MX will show you the MX records for the site, or dig sitename.com A will provide the A record(s).