For the above example, you definitely want indexes on the group and SSN columns.
To get the total row count of matches, use COUNT():
$sql = SELECT COUNT(*) FROM `t` WHERE `group`= '2' AND `SSN` LIKE '%1212%';
This will return just one result row with one column (the count) as opposed to returning every column of every matching row.
Finally, on your actual data query (with the LIMIT clause), unless you really need all 600(!) columns in each result row, instead of doing SELECT *, provide a list of only the columns you actually need. (Yes, it may be a lot of typing if you need 60 of them, but that would still mean transmitting only 60 columns of data per result row instead of 600, or roughly 10% as much (obviously varying depending on the relative sizes of the included/excluded columns).