When playing SMT:
1. Never get "close" to a demon. Always make sure your demon party is up to task depending on the floor, or even dungeon. In order to gain new abilities and demons (persona, in this case) you must fuse.
2. The logic between fusion is that each demon has its own unique type or 'race'. In SMT proper, Angel is an Divine type demon, for the purpose of Persona series, demons which are called Persona, have Arcana. In Persona, Angel is a Justice type Persona. In order to make Angel you need x level of two different Justice Personae.
3. To create a specific demon, you must know fulfill level requirements. You will know what the level requirements are by looking at the demon in question, if it is blurred out, and then see what level is required. The game will tell you.
4. Each demon has a specific ingredient, i.e. this demon, or that demon, to make the persona. Some don't have heavy requirements, but if you want to make a Metatron or extremely high level demon, you are going to have to go through a specific amount of steps. Sometimes on rare occasions the fusion WILL mess up so always save before you fuse.
Addendum (basics of playing SMT):
Cards contain persona and each dungeon has a specific set of demons you fight. Depending on the situation you will need to switch up your persona roster. If the majority of the demons are weak to wind, get a wind based Persona or two. Always cover your bases, so that you have Persona that protect from whatever strengths the enemies throw at you, this includes death magic.
It's suggested to try to level up all slinks, even with characters who may not interest you because this increases their potency in battle as well as leveling up your Arcana types. For example, Chie's arcana is The Chariot. By leveling up Chie's slink, you are by leveling up Chariot. This means you can get better Chariot Persona. By balancing your slinks, you are by also helping yourself to a wider palette of peronsa to fuse and create.
Fusion is the hallmark of SMT demon collection. You fuse to make whatever demon you need at the time, or that you think you will need eventually. Any Persona you make is entered into your compendium, which allows you to buy and rebuy any demons you discard or fuse. Need to make something that requires your Jack Frost? No biggie, save your current Jack Frost so you retain levels, spells, abilities in the Compendium, then save the game, fuse, go to compendium history, re-order that Jack Frost, you got your persona back for a select fee. The more powerful the Persona the more it costs to get them back from the compendium.
With S links, compendiums, fusion, and roulette card shuffle, you have a lot of options to get ahead in P3/4.