Though this is not a MTB quest persay I thought you guys would have the info.
I am building up a 2 Surly LHT touring bikes and will be using a flat handle bar. I want to go with trigger shifters. On our Giant Escape hybrids we have a shifter/brake combo. I see that I can use seperate shifters and levers. Seems like the way to go? I was thinking maybe Deore?
Would appreciate your advice!

Shimano or SRAM..

On a Touring bike I'd use Shimano even though I prefer SRAM because Shimano is everywhere.

Shimano

Use seperate brake levers and rapid fire shifters, that way if the brake levers are damaged, you don't have to pay for both brake and shifter replacement. The seperate units also means that if you were to go the disk brake direction later, you do not have to change the shifters.

Being realistic, unless you are talking budget brake lever/shifter combos. the format is dead.

Shimano hasn't made MTB STI's since the ill-fated DCL's, and SRAM never has.

What you do get now is matchmakers/I-Spec, which allow you to use the same clamp for the brake lever & shifter, some aftermarket brands even produce their own to allow aftermarket brakes with SRAM/Shimano shifters.

For which to choose between SRAM & Shimano, this is governed by your derailleurs, as normally these need to match the shifters.