When you got your ultimate up or at least your teleportation up -> you go push the top or the bot lane.
When you don't have your ultimate or teleportation up -> you try to stay a bit close to them in case the ennemy team wants to fight.
The fact that Shen can instantly teleport on one of his ally if a fight is about to begin or was engaged makes him the perfect champion to splitpush. You will have to keep pushing until someone on their team is going to defend the lane your pushing, you have to keep a constant eye on your team mates because the ennemy team can decide, instead of going after you, to engage the fight because it would be a 5v4 for them. This is when you have to use your ultimate and join the fight.
You also have to use your ultimate on the right person, here are some advices that can help you :
AD Carry / Mid laner : If you see one of your carry being focused by the ennemy team, you use your ultimate on him, if he's safe during the fight you can save your ultimate for someone else.
Jungler / Support : If your jungler/support is a champion like Shyvana/Thresh who is strong to engage or just go in, you can use your ultimate on him to follow his call, by doing this you will prevent some ennemy champions to go on the team fight.
A simple way to use your ultimate is to see who is being focused or who is pretty low life and then just use your ultimate on him, plus if your mate is being focus it means that there are ennemy champions close to him so it gives you a better positioning to taunt.

I'll complete this guide when i'll have time, for now i think i gave you the essential to play an effective splitpush Shen (in my opinion).