When you synthesise a soldier card with another soldier card, the base card gets experience.
When you synthesise a soldier card with a card that has the same soldier ability as it, the base card gains experience and there is also a chance that the base card's soldier ability will improve.
When you synthesise an instructor card with a soldier card, the base card gets a lot of experience and there is also a high chance that the base card's soldier ability will improve.
Choose the third option to use "rebirth" - you can only do this with a card that has reached its max level. Different cards require a different weapon to use rebirth. After using this, the card's level goes back to 1 and you can level it up again to increase the health and ability gauge refil speed even further.
Choose the fourth option to remove the card from the squad.
When you press triangle while in the squad organisation screen, you get a list of all your cards. Select one and choose the 4th option to remove the soldier from the 3rd Shinsengumi unit (when you want to clear some space for other cards, or have no use for a particular card)