You have the pipe the other way around and you need to echo the query, like this:
myvariable=$(echo "SELECT A, B, C FROM table_a" | mysql db -u $user -p $password)
Another alternative is to use only the mysql client, like this
myvariable=$(mysql db -u $user -p $password -se "SELECT A, B, C FROM table_a")
(-s is required to avoid the ASCII-art)
Now, BASH isn't the most appropriate language to handle this type of scenarios, especially handling strings and splitting SQL results and the like. You have to work a lot to get things that would be very, very simple in Perl, Python or PHP.
For example, how will you get each of A, B and C on their own variable? It's certainly doable, but if you do not understand pipes and echo (very basic shell stuff), it will not be an easy task for you to do, so if at all possible I'd use a better suited language.