Lots of options, almost all do-able.
But, going from 6-sp to 9-sp means you're potentially adding 50% more shifting to your routine.
There's a slight learning curve, because the shifters are light and you can feel ham-fisted in a hurry.

I stepped from 6-sp friction to 9-sp friction with just a wheel change. Boy, that took a touch...

I went from 7-sp indexed to 8-sp DA indexed, on the DT. I kept the range, loved it.
Then, I figured if 8 was great, 10 was better, and bought some DA7800 DT shifters, RD, cassette.
I was very proud of myself for doing the upgrade without spending so much on STI shifters.
The moment I realized I had all those gears, I must have forgotten how to ride 7 or 8.
I complained about having to shift so often, without realizing I didn't have to.
My riding bud had the right comment. "Ride like you still have seven, and shut up."