If I want to buy crypto, then I believe I have to go to a broker or an exchange (assuming I don't know someone who wants to sell me some). So I've got to pay them. Are they regulated enough that I can be confident that they won't just run off with my money once I've paid them?
The exchanges are not regulated but they are centralized which makes it similar to been regulated since it is controlled by a body. You can buy from them but do not leave it on there because they could encounter any problem like hack or something similar which will affect your funds saved there. Can I pay be credit card? Then, once I have bought from the broker or exchange, do they transfer the crypto to a wallet that I control? Or do they keep the crypto in their own wallet as a custodian? If so, again, how can I trust them?
Yes some Exchanges support the use of credit card but it depends on your country’s restrictions on cryptocurrency, if you can’t use the credit card then you have to use the P2P method. They do not transfer to your own wallet but rather they keep it in a custodian wallet they created for you on the exchange. So you can withdraw to your preferred cold wallet which is the best to do.
Also, is there an advantage to buying from a broker over buying on an exchange?
It is almost same if it is not a P2P transaction on a centralized exchange. It is more secure to buy it physically from someone known but this could also be risky in a country where cryptocurrency is banned.
