How to collect the logs from multiple machines to my server efficiently?

You don't need to start filebeats in any particular sequence, but logstash should be running for the filebeats to be able to ship the logs.

In logstash you need to configure the input plugin for beats, and in beats the logstash output.

Once everything is configured, if it doesn't work you can check the logs of filebeats to see if they are reporting any problem.