I trade on at least six exchanges regularly. The most popular are Poloniex and Bittrex, because of high trade volume, and they are safety enough...but...there are lots of coins that are no trated on these exchanges, moreover, they often have problems with transactions and wallets because of lots of new people coming into the market. You can also look at HitBTC and Binance

By the way I would recommend to look closely at decentralized exchanges, I thinkg they will gain more popularity due to more regulation laws in many countries. BUT whatever exchange you choose...don't keep all your money there, no one knows, even such big exchanges like Poloniex and Bittrex can be closed or compromised, and if you read their policy carefully you will see that none of the exchanges takes responsibility for your funds. .