Coinbase is about a legit as they come. They have some reported customer service issues, but you have to remember that they are a startup, with all the ugly warts that those companies can come with.
However, there are some alternative exchanges. GDAX and Bitstamp also offer Fiat to Cryptocurrency services, at least for BTC. If you want to move into Litecoin, you would need to go one step further with a service like Changelly or Shapeshift. There you can exchange into almost any popular altcoin.