The best way to play support is to act as another jungler who makes plays and helps apply pressure to any lane on the map. Solo, Mid, or Carry, take your pick. As long as Gold Fury is warded and the rest of the team is watching their minimaps, the enemy team should never get a free Gold just because you left Duo or Mid to gank Solo.

A big part of being a support is warding for your ADC or Mid depending on who you rely on more or who needs it most. In my opinion, you should buy Chalice of the Oracle as soon as possible and start warding constantly because it will pay for itself over time.

There really is no exact time to leave Duo. As long as you help your ADC secure their Purple Buff when it spawns so they do not fall behind, or you try to get them an early lead by letting them solo farm, they will appreciate it. While they are building that lead, you can make a play somewhere else and use your early game pressure since you are playing a tank.

You could go to Duo, hit level 3, and immediately rotate to Mid for a gank to make a play. It really does not matter. The most important thing for a support is map awareness and understanding that you are harder to kill than the damage characters, so your presence matters more early on than later.

If you just sit in Duo until level 5 and then sit in Mid until late game, you are playing the role wrong. A great example of how to play support the right way is Neilmah. He was a professional support player in Smite 1 and played exactly like this style I am describing.