Agreed that you should look at the verification process of the exchange.

More fundamentally, you will want to make sure that it is reputable (ie is not a scam), that there is decent volume being traded in the coins you wish to buy/sell and that the exchange offers you sufficient security options (MFA/2FA is a must).

In general, one way to identify the best exchange for you (you can use more than one FYI) is to look at the coins you want to buy/sell on CoinMarketCap.com and see which markets they are being traded on. Once you know where those coins are trading with the highest volume, you should have a better idea of which exchange you would prefer to use.