It is always a good idea to go for 40% (or 45%) CDR on Shen.
Tenacity is desireable, as you cant get Unflinching in your runes.
HP is the primary stat that scales with your kit as support.

Choose the Items you build according to the enemy team and your own team:
Magic DPS -> Adaptive Helmet
Healing/Autoattacks -> Thornmail
Poke -> Warmogs
Crit -> Randuins Omen
Healing on your team -> Spirit Visage
High burst -> Gargoyle Stoneplate
Assassins -> Locket of the Iron Solari

Situationals:
Abyssal Mask: When your team has high magic damage threats and the enemies are tanky.
Zekes Convergence: When your team has a fed ADC and you stay near him and ult him every fight.
Zhonyas Hourglass: When there is a fed enemy AD-Assassin and you have Guardian/Relic Shield
Redemption: Last item choice for teamfighting in the "super lategame"
Mikaels Crucible: Last item choice if a carry that doesnt have Quicksilver Sash against a CC-threat
Frozen Mallet: Replacement for Pauldrons in the final teamfight of a game ("super lategame")

The starting items as support shen are either Steel Shoulderguards or Relic Shield.
On the first back you want to start building the Knight's Vow and Boots.
Priority is a Kindlegem then the Boots and lastly the Chainvest.
The next back after this you ideally have enough gold for the rest of the items metioned above and additionally enough gold for a Cloth Armor or Null-Magic Mantle to build into your finished boots later. If you can finish the Knight's Vow, do that before working on your Boots.
After that, you should buy another Kindlegem, it is recommended to always buy one after using it for an item, since you want 40% CDR and 3 core items (Knight's Vow, Spirit Visage, Warmog's) build out of it. This includes the cases where you dont buy an item that uses a Kidlegem after the Knight's Vow. You can just sit on it until you need it, since the HP is a very good stat for you. The Items you build rely heavily on the game's development, thus there is no real "go-to" build order after these general shopping rules for Shen.

Shen's support playstyle varies over the coures of the game because his roles change.

Pre level 6:
Look for favorable engages on the squishier immobile target on the enemy botlane to try and get a few early kills with your damage and CC.
After level 6:
Now its safe to roam and try to gank midlane, because you can ult your ADC if he gets engaged 1vs2. The other option is to look for ultimates toplane or midlane (or jungle) when your ADC is recalling to buy Items. This means you dont leave him alone on botlane for too long, since you can recall after the ultimate and get to your lane not long after your ADC arrives there.

After the laning phase:
Peeling carries:
You basically follow your most fed carry around (either your ADC or your midlane Mage) and protect them from any threat. This includes teamfighting.
Frontlining:
When you are the only Tank and Engage, you look to threaten the most dangerous enemy in teamfights and try to draw most of the attention to you, so your carries can kill the enemy team.
Splitpushing:
Very situational, but you can defend a splitpush against a not too fed enemy melee champion. You are too tanky for them to kill you just like that and you can prevent them from attacking the turret you defend with the threat of forcing them to dive you due to your taunt. Not the most effective defense against a splitpush but enough if noone else on your team has teleport up and the enemy is looking for a Baron.

Shen's Ultimate is a very powerful spell and thus has a long cooldown.
Your goal is to use your Ultimate as best as you can to get and keep your teammates ahead of their enemies.
Ulting a teammate to safe their live if they have a shutdown even without getting anything else is worth it.
Ulting a teammate for a dive or to turn a gank and get them kills is worth aswell.
Ulting your ADC in a 2v2 on botlane after you used all your spells is also a good way to make him survive and win the fight.
Ulting away from your ADC on botlane and letting him die is not a good idea, so Ping your Play beforehand.

Ulting in teamfights:
When ther is a champ with better engage than you on your team, ult him when he is about to go in. That makes you pop up in the enemy ranks and you can follow up witho taunt and Spirit Refuge to break their formation and simply lock them downt for your carries to kill.
When a carry is about to die and you didnt ult at the beginning of the fight, sacrifice yourself and ult him even tho that renders you unable to do anything for 3 seconds and you will probably die.
When you are currently not in the teamfight and ult to join in, it is sometimes better to ult a tanky teammate to guarantee that you will blink there. Sometimes your Carries get oneshot and then the Blink from your ult cancels, levaing you unable to join in.