Your steps should be something like this:
Variant 1 - if you can't access the database directly over the network:
0. Connect to server via ssh
1. mysqldump > /path/to/bakcup/file.sql
2. Presume you copy over ssh: scp user@server:/path/to/backup/file.sql /path/to/local/file.sql
3. you don't have to search for the db file, since you know where you saved it. I you want to search inside the db file for a particular string, you can do that with a lot of tools. As suggested above, grep is a good choice.
Variant 2 - if you can access the database on the server over the network:
1. Be sure you have mysql-client installed on your computer (there are versions both for Linux and Windows PCs)
2. use mysql dump (read the manual for exact parameters) to dump the file over the network directly to your computer