How do I level up my partners and leader quickly?

Leveling up your partners is really easy. You just need to keep feeding them with Nightmare (EXP) bottles. Please follow this link to learn how to farm these EXP bottles. 

https://www.appgamer.com/illusion-connect/strategy-guide/how-to-farm-nightmare-bottles

However, for the Leader, his EXP accumulation will be different and slower. You can only earn EXP by completing stages. Thankfully, raids and repeating previously-cleared stages still give your Leader EXP so basically, just clear stages and dreams regularly and frequently.