Why does it say first backup?
Because this is the first time that Time Machine has backed up your Mac to the G-Drive.
Do I have to keep it connected for more backups?
Well, the G-Drive has to be connected to your Mac if you want to use Time Machine for subsequent backups.
Ideally, you would leave the G-Drive connected all the time and Time Machine will backup automatically after you have been using the Mac for a hour or more. In other words, you don't have to do anything for backups to occur automatically.
If you don't want to leave the G-Drive connected all the time, maybe you could connect it for a hour or two each evening, so you will have an up to date backup of your Mac each day.
Not recommended, but some users only back up once a week or so. The downside.......say you back up only on Sundays and your Mac's hard drive fails the following Friday. You have no backups for Monday, Tuesday, Wednesday and Thursday, so that data cannot be restored once your Mac has been repaired.