Absolutely right to feel kind of 'awkward', you have good senses. If your going to use an exchange try to stick with the more reputable (i.e. well known ones), bitstamp, btcchina, coindesk, etc. but as others have already mentioned, don't keep any of your long-term bitcoin savings on an exchange. Your basically trusting the keys to your crypto-wealth to another party when you do and any number of things can obviously go wrong in that scenario.