Totally understandable - better safe than sorry, especially while learning.
Yes, the batteries, remote, and aircraft all receive separate firmware updates that may or may not be released simultaneously from DJI (i.e. there may be an aircraft update, but no updates to the remote or batteries at that time).
The one you really need to watch out for are the battery firmware updates - most people have the Fly More combo and therefore have at least 3 batteries. If there is a battery FW update, you will need to update each battery separately and individually by shutting down the aircraft, swapping batteries, and turning everything back on again. That will force a firmware check through the DJI GO app, and it will prompt you to update the battery each time. This can all be done wirelessly in your home, no need to hook it up to the computer or anything, but you will need everything hooked up almost as if you were ready to fly. It's not difficult, it just takes a few extra minutes and you need to remember to do it