I personally prefer to choose exchanges with more lax verification or none at all, which led me to largely use BitMEX and Binance until the former stopped letting American customers trade on their platform. Having a somewhat longstanding reputation and decent volume is something else I like to see in an exchange, as I don't trust exchanges that are exceptionally new or have low volume.

Good support and quick transactions are also things I look for in an exchange. Errors do occasionally pop up for any service, and when they do, it's important that you can get assistance as quickly as possible for an issue.