Change out shifters to be push-pull

I'm presuming you mean any shifter with a downshift thumb button and a trigger type index finger upshift release, i.e. most Shimano shift pods.

The hoop you must jump through here is that there aren't tons that are 2x8 by design. Most are 3x. Running a 3x shifter on a 2x front crank can be made to work acceptably, and many bikes have been set up that way, but getting it dialed great so that "shift lock" situations are properly unattainable can be a little tricky. Given that you have the option, I think just getting a native 2x shifter is better. One option for this would be the SL-M315 set from Shimano. That's the only set I'm aware of that has the clicks you're looking for and is a matched set.