When people said don't trust exchanges, they didn't mean that you shouldn't use them. They meant that you shouldn't store your coins for long term in any exchange. Do your exchange/trading and then when you are done trading, move your coins as soon as possible. Don't use those exchange wallet as long term storage wallet. Bad idea. Exchanges might get hacked or your account might get compromised and you will end up losing everything.
As for webwallet, just avoid those at all cost. These are the least secured wallet where you can store your coins.