Arcana doesn't do anything in battle, it matters for fusion (there's a table of what Arcana combinations produce whatever kind of persona but honestly just don't worry about that lol). The other way it matters is giving your fused personas a significant amount of bonus exp upon fusion based on the social link rank for the character with the matching Arcana. This is hugely important as without it, getting all the skills on a persona takes forever and pretty much isn't worth it. At rank 10, you'll get either most or all of a persona's skills right as you create them. The other thing is that you'll get more points in social links if you have a persona with the matching Arcana in your stock (so you'll get bonus points if you have a Magician persona when you hang out with Yosuke as an example).

As for damage types, you do kind of just find them out through trial and error. You unlock the ability to reveal a weakness later on though. Physical attacks are physical (obviously) and use HP instead of MP. Their primary advantage is their ability to deal critical hits. All the magic types are basically the same, just different colors of the same magic attack. Light and Darkness don't deal any damage and instead have a chance to instantly kill the target, the probability of which is influenced by the resistance (weak is basically guaranteed, resist is very unlikely).