MySQL 5 - How to check for max number and change max number of connections?

Question

Am using legacy Spring 4 app which uses MySQL5 on the backend.

When using raw Spring JDBC calls, a bunch of DAOs are throwing the following exception:

DAO::org.springframework.jdbc.CannotGetJdbcConnectionException: 
Could not get JDBC Connection; nested exception is

com.mysql.jdbc.exceptions.jdbc4.MySQLNonTransientConnectionException:

Data source rejected establishment of connection, message from server: 
"Too many connections"

After scouring the web using Google, I get a lot of different choices for solutions so I rather just ask here.

Tried troubleshooting using the following:

When I issue the following commands inside the database:

    show status like 'Conn%';

Yields:

Connection_errors_accept            0
Connection_errors_internal          0
Connection_errors_max_connections   395711
Connection_errors_peer_address      2
Connection_errors_select            0
Connection_errors_tcpwrap           0
Connections                         423604612

A different SQL call:

show status like '%onn%';

Yields:

Aborted_connects                    6857
Connection_errors_accept            0
Connection_errors_internal          0
Connection_errors_max_connections   395711
Connection_errors_peer_address      2
Connection_errors_select            0
Connection_errors_tcpwrap           0
Connections                         423605165
Max_used_connections                1001
Performance_schema_session_connect_attrs_lost   0
Ssl_client_connects                 0
Ssl_connect_renegotiates            0
Ssl_finished_connects               0
Threads_connected                   12

Question(s):

1. What is the max amount of connections that can be set by MySQL without it causing any breakage or further performance issues?

2. Checked for max connections like this:

   show variables like "max_connections";

Yields:

max_connections 1000

A lot of articles on the Internet states that I need to grant / edit the max_connections and max_user_connections in my.cnf file under the database server. Also, read that I could something like this:

mysql> SET GLOBAL max_connections = 3000;

What is the best course of action to take regarding this issue?

Is it by just simply grant / edit the max_connections and max_user_connections in my.cnf file under the database server as root? Or, are more steps needed?