I totally made a flow chart so I could get every support convo in the game 8D
As for game play pointers. If you play with a cleric/healer of any kind, Get them to a promotable state as quickly as possible. I hate having Characters attacked when they can't defend or attack back.
If you're not on lunatic, spamming your rescue or other high ranking staves will help with this immensely. Another cheap trick is to go to a map where you block an entrance point with a character that can be hit and damaged but doesn't die, take away their weapons and spam heal. Mend or psychic if you want more efficient but expensive use of turns. As long as no other enemy in range and they don't have to retreat due to health loss, the AI will keep attacking your meatshield. A good, early map for this is the one castle where you get Sumia and Kellam.
Alternatively, because the above is kind of cheating, if you don't like pure healers swap them out once you get access to promotions. Thief to trickster gets staves, Pegasus Knight to Falcon Knight gets staves and mage to Sage gets staves. Gaius isn't really worth crap as a healer until he gets access to the Recover Stave because of his low magic. Sumia should make a decent healer though, and your mages obviously have the stats to do the job as well.
As for getting all supports: play casual. Save before achieving an S rank. Get S rank. Go fight. Make a battle save. Reset. Open the save before the battle. Repeat with other partners.
Unlimited supports are so exploitable.