Coinbase is one of the most popular and trustworthy Bitcoin exchange companies.
Like any bank, Coinbase is FDIC insured up to $250,000, and must comply to typical audits and reporting standards.
You can buy bitcoin with a Bank account, debit card,. or credit card. Coinbase also allows you to send and receive purchased digital currencies, providing you with the appropriate wallet address.
This video from one of my favorite youtubers does a great job of explaining the process: