Your error typically means that a firewall blocked your connection attempt.
For Cloud SQL, this typically means that you have not enabled instance-level access.
Since your error is 10060, which is a Windows WinSock error message, I am assuming that you are running on Windows. The steps are similar for Linux.
There are two methods to enable access to Cloud SQL:
Cloud SQL Proxy
This is the recommended method and is secure. Download and install the Cloud SQL Proxy to your the computer that you want to connect from. If you have set up the Cloud SDK (gcloud), Cloud SQL Proxy will use your Cloud SDK credentials. If not, then you will need to create and download a service account with (usually) the Cloud SQL Client role.
You will need the "instance connection name" from Cloud SQL Instance details page.
- Download Cloud SQL Proxy (link) to your computer. Remember the full path to the program. Save the program as "cloud_sql_proxy.exe"
- Start the Cloud SQL Proxy
[full_path_to_program]\cloud_sql_proxy -instances=[INSTANCE_CONNECTION_NAME]=tcp:3306
OR
[full_path_to_program]\cloud_sql_proxy -instances=<INSTANCE_CONNECTION_NAME>=tcp:3306 -credential_file=[PATH_TO_KEY_FILE]
- Connect using mysql.exe: `mysql -u USERNAME -p --host 127.0.0.1
Public IP Method
The second method is not secure unless you also enable SSL, which I will not cover in this answer. I do not recommend this method except for knowledgeable persons.
Note: Most home Internet services change your Public IP address periodically. This means that you will need to update your Public IP address in Cloud SQL if you are using the second method.