The exchanges I am currently using are Bittrex, Kraken and Gdax/coinbase. I like bittrex a lot because they have so many coins. This exchange is a must. Kraken has about 10-15 coins available and coinbase/gdax has only 3 coins(BTC, LTC and ETH).
Also there are smaller exchanges like Cryptopia and Novaexchange where you can get lesser known coins. I have used these exchanges without any problem as well. Binance is another exchange which has its own cryptocurrency BNB.
Hope this helps.