@the1up, you didn’t mention the Hat Shop so I won’t assume you’re familiar with it. There you can scroll down the list of hats that are available for sale, what they cost, and what they’re worth in terms of power. It will also indicate whether you already own a particular hat.
Some “events” feature hats that can only be acquired through the tower. Typically you will need a lot of gems to get many of them that way. After the event is over, you can buy any remaining hats from the hat shop.
If you’re new to collecting hats, avoid my rookie mistake of buying them randomly or because I just liked them. Start with the cheapest hats and don’t move on to the next hat type until you’ve completed the set and gotten the slingshot upgrade. The reason: a set of cheap hats deliver the same slingshot upgrade power as the more expensive ones, even though the individual hats themselves may be worth less.
Also once you have a set of, say, “common” hats, it doesn’t help you to buy just 1 or 2 from another common series, since you’ve already got the max fire power you can get from that level of hat. Wait until you have enough pearls to buy the whole set. The reason: you’ll pick up a lot of cheapo hats during normal game play and it doesn’t make sense to waste pearls.
Pardon me if you know this already, but you did say you were new….