There are lots of secure sites and wallets that let their users buy bitcoin right from the wallet. There is Coinbase, one of the best wallets and exchanges linked together. You can buy bitcoins through the Coinbase app or site with your debit card and it's fast and secure. As a matter of fact, Coinbase is being used by most starters and that's the first I used.
 
Another good one is the Blockchain wallet. Recently they started allowing their users to buy Bitcoin from their wallet and they also added an exchange so that users can exchange any of the available coins. You can also make use of the Binance, Huobi, OkeX, and LocalBitcoin (decentralized exchange), and Paxful.