Support ADC synergy:
In general passive support should go with passive adc. Aggressive support should go with aggressive adc.
If your adc is a lane bully like lucian or draven a support that has cc and can go all is good. Leona and zyra are example. If the adc just wants to farm like ez or kog or jinx then a support with peel or heal is good. For example braum or janna or soraka. Support with poke is good with any adc. Many adc are strong early game and game and can more or less 1v2 (mf and cait) they can go with any support. IMO these are the best adc if your support is garbage. Some supports have a bit of everything poke peel, cc, shield, (nami, sona, thresh) they can go with any adc. In general ranged support are stronger in lane since they have better poke, melee support are better with all in and team fights.
Push or freeze lane:
In general push the wave to get map control, freeze the wave to deny farm.
Push first wave to get level 2. Push the wave to their tower if you have enough gold to get the item you want and want to recall. This gives you time to walk back to lane without missing cs. Push the wave if your jungler is invading or a fight is about to happen in river or jungle, this prevents the enemy bot lanner from helping. Push the wave if your jungler wants to sneak dragon. Push the wave if you want to relieve pressure from the other lanes and you have good escapes (this will make the enemy jungler gank you rather than mid or top). Freeze the wave near your tower if you are stronger than the other bot laner to stop them from farming. Freeze the wave near your tower if you jungler is going to gank soon.
Farm or fight:
In general fight if the enemy is out of position or if you have a number advantage (jungler gank) or item/ level advantage.
Fight if you reach level 2 first. Fight if your minion wave is bigger than theirs. Fight if they are out of position or if you are stronger want want to contest farm. Fight if the enemy is trying to cs. Fight if the enemy skills are on cd. Farm if the enemy minion wave is larger (This is pretty important in lanning phase). Farm if there is no team fight or objective going on in the map.
What to do if you killed enemy laner:
Try to get advantage after the lanner died. Take the objective that you think you can get. Ideally dragon or tower. If not then at least push in the minion wave or pressure another lane/ jungle.
When to bacK:
ideal situation is when the adc has enough gold to get a good item buy or one of the enemy laner was killed. adc and supp push out the wave and back together. If one of you got killed or you are pushed under turret then you should stagger the back to have one person hold the turret.
Who to focus in fight:
Focus the adc in most cases since the adc does more damage. IN certain situations you should focus the support if the support can keep the adc alive (soraka or tham). If you can only get one of them kill the one that you can instantly pop (ex if support is at 25% health kill the support)
What to do if you are pushed under tower:
Very few adc can farm well under tower (sivir jinx kennen). Ask jungler for a gank if this happens. As the support you should hit each ranged minion once so that it takes 1 hit to kill after tower. Or else the ADC will miss some of those.
Warding:
Ward the area you think you'll get ganked from. It's usually the bush closer to where the minon wave is. Ward dargon if the enemy have a jungler that can easily solo dragon like shyvana ww or tyrnd. The first gank usually comes around 3:30 mark. However junglers usually start bot buff and gank mid or top first.
When the group:
Group once your bot tower or enemy bot tower falls. If enemy bot tower falls, push in the wave and go help mid. If your mid laner is a mobile champion or a good skirmisher (zed le blanc) then the adc and support should seige mid and let the mid lanner hold bot. If the mid laner is not one of those champions then there's 2 things you can do. 1. use the support to help mid laner seige mid and the adc hold the tower (freeze wave dont push). 2. supp stay with adc push out bot wave go seige mid. once the bot wave goes back to your tower supp and adc return to bot. If you lose bot tower first then 1 person hold bot at inner tower (usually adc can be the mid laner), have 2 ppl seige mid (usually supp and mid can be supp with adc). Once more towers fall and you are winning rotate around map have 1 person push out a lane and join the group. If you are losing, then farm at towers to stall game.