Are there specific wallets I should be using? Should I keep my Bitcoin in an exchange wallet, or is it better to use something like a hardware wallet?
There are lots of quality and safe wallets to use but not any of the centralized wallets as they're becoming a norm that they even get suggested to newbies. Hardware wallet are the ideal wallet but as a newbie I won't recommend you start with that to avoid making some mistakes that causes you, your coins in the future. We have few wallets like electrum that are good software wallets to use. Exchange should only be for trading and not hodling of your investments. Before investing, invest your time on researching to know more about the industry to avoid regrets because there's no refunding or recovery of stolen coins in the industry.