I take it there will be a guide on how to obtain all companions later on.
I'll write down what I know off the top of my head. These are the methods to unlock the Touching Minigame for girls, so that you can unlock them as companions:
Imp Girl - You must be level 1
Succubus - Defeat a Succubus in Battle. An NPC will appear two maps below the first town. She will ask you to cook her fried onions. Do so, and give it to her.
High Succubus - Defeat her once. Unlock 10% Physical Strength Passive Trait in Skill tree. You will find her mini game hidden in a secret place. Go one map above the first town, and check the wall to the left.
Sea Succubus - Defeat her with her weakness
Downer Succubus - Scan her in battle, and then defeat her.
Butterfly Girl - Defeat her once. In the cave you'll find one blocking a path to a room. She'll ask for food (I forget which but you can buy it.) Give it to her.
Door Mouse - Get her to put you in a Hold position, and let her breastfeed you for 10 straight rounds without succumbing to her. Then defeat her.
Goddess - In the mansion there is a succubus that requests 70 Angels Feathers. Collect them off floor and give to her. She'll give you a Rosary. Defeat the Goddess and her minigame will unlock.
Pirate - Defeat her once, and have 50,000 gold in inventory.
Red Haired succubus (forget name) - Defeat her once. Have Strawberry milk in inventory.
Cowgirl - Same exact method as Doormouse.
Slime Girl - There's one you can talk to in Submarine tunnel. Have mucas in inventory and she'll join you.
Club Knight - See below
Heart Knight - See below
Spade Knight - See below
Diamond Knight - You must let them level drain you 30 times, or defeat them with a 3% chance droprate of dropping a key that allows you to open a chest in the castle with an item that unlocks minigame.
Carol - She'll automatically join you in the Submarine Tunnel
These are the ones I know for sure.
The Knights in the castle are the most tedious. If you want to go for the Level drain method, the best way to do it is to let them get you in a hold position. They will level drain you every round. So make sure you're at least level 31.