Confirm MySQL version

$ mysql -u root -p

mysql> SELECT @@VERSION
+-------------------------+
| @@VERSION               |
+-------------------------+
| 5.5.60-0ubuntu0.14.04.1 |
+-------------------------+
1 row in set (0.00 sec)

Take a backup of all databases

$ sudo apt-get update
$ sudo apt-get upgrade
$ sudo apt-get install mysql-server-5.6

Login to MySQL and confirm version.

$ mysql -u root -p

mysql> SELECT @@VERSION
+-------------------------+
| @@VERSION               |
+-------------------------+
| 5.6.33-0ubuntu0.14.04.1 |
+-------------------------+
1 row in set (0.00 sec)

List databases

mysql> show databases;