I'd love to hear how this works.
So basically I play Shoshone and beeline Military Tradition first. This gets me access to Barracks pretty early on, and therefore two sources of faith per city (shrine and barracks). Military tradition is also where the Shoshone UI unlocks, which I find to be rather powerful, plus you can built horsemen! Because Animal Husbandry is a prereq for MT, I know where the horses are on the map early-on.
In terms of build order, I start with a scout, then go shrine, warrior, monument, warrior/archer -> Temple of Artemis -> Settler, Archer, Settler, Barracks, Horseman. ToA is not really optimal, I just like it. I tweak this order depending on context of course. I use the Progress tree to speed up my tech aquisition and give me a free worker to build my UI and grab those horses.
The other thing that's relevant is that I use my scouts and the Shoshone's ability to pick Ruins upgrades to get a pantheon as soon as it is available! After Military Theory I beeline Sailing so that my recon units can grab those ruins I spotted off the coast (as well as discover city states and new landmasses) before anyone else does.
During this process I can usually get two faith boosts from ancient ruins with the help of my (now fairly high level) recon units
. I'm usually also able to get two free techs during my early exploration too (most commonly is one of them is Pottery or Trapping, which are very useful). For context, I play on Prince, Epic speed, Huge map, Continents Plus.