You should always have your own local wallet, and if you use an exchange, you should immediately transfer your funds to your local wallet once you buy them.

In terms of exchanges, there are many alternatives to Coinbase which you should look into. Coinbase has some sketchy history like insider trading, blocking Wikileaks without giving reasonable explanation, several times they've turned off their markets when conditions aren't favorable to them (sudden price drops). They're also not able to keep up with the technology as they've lagged way behind many of the other exchanges when it comes to implementing technical updates like segwit, bech32 addresses, transaction batching. God only knows how long it will take them (if ever) to be ready for the Lightning Network.