Its not one or the other, its just whether or not you want to learn manuals.
If you learn to drivea manual transmission, you'll have no problem switching to an automatic. I own 1 manual car and 2 automatic vehicles, and switch back and forth all the time.
Manuals are pretty rare these days, so if you simply don't want to bother with shifting then its perfectly okay to just learn to drive an automatic and never bother learning stick shift. You'll just need to make sure every car you purchase is an automatic, but that's not hard since most cars are. If you need to rent a car pretty much all rentals will be automatic. Most rental places don't even offer stick shifts and the few places that have them will also have automatics.
So really you only need to learn a stick shift if you want to, or if you feel like its a life skill you want to have just in case.
I've had several situations where I was unexpectedly asked to drive a stick shift. The most notable was in college when I went snowboarding with some classmates. The guy who drove our group to the ski area broke his wrist that day and the vehicle was a stick shift. Fortunately I knew how to drive a stick, so I was able to drive our group back to town and drop him off at the hospital.
But that was 15+ years ago when stick shifts were still somewhat common.
The bottom line is that you only need to learn to drive a manual transmission if you really want to.