For most people in the U.S., Coinbase would be the easiest option to buy Ethereum, Bitcoin, or Litecoin (it doesn't support any others yet). After verifying your account, you can add a number of payment methods including credit or debit cards, U.S. bank accounts, or even wire transfers of funds. Other options for exchanges that will take U.S. dollars for coins are Kraken, and Gemini in the U.S. Typically you will need to verify your account with a driver's license and add other details to expand your buy limits. Since cryptocurrencies are "hard currencies," the exchanges don't want to risk getting ripped off, since you can't reverse a cryptocurrency transaction once it's done.
If you are looking for some of the newer coins like NEO that are making big movement but haven't made their way to the aforementioned exchange sites, you can look into Bittrex, Poloniex or Livecoin. You can transfer Bitcoin or Ethereum to these platforms from Coinbase and then exchange it for any other digital currency that you want.
If you are outside the U.S., here are a few options for exchanges that take your local currency: BTC Markets (Australia), Bitthumb or Coinone (Korea), CHBTC or Huobi (China), and QuadrigaCX (Canada.) You can find a full list on this page of where to buy Ethereum for your local currency.