I just got this question so I thought it is a nice opportunity to post about it right away.
For starters there are a lot of different ways to buy cryptocurrencies. One of my favorite at the moment is LiteBit.eu. The reason why is that you can buy currencies directly with IDEAL so you don’t need a creditcard and you also do not have to buy Bitcoins or Litecoins first. A lot of people in Europe have found this site. In 2 weeks this site has extremely more traffic than any other time.
I think there will be more sites like this but for now this one is my favorite.
A lot of people also use Bitrex, Binance or Coinbase. Coinbase is easy to use but has no IDEAL payments. Bitrex and Binance I would not recommend right away for starters, but if you have a bit of experience in cryptocurrencies there are a good place to go.
All of them has online wallets , but remember anything can happen so it is wise to transfer your currencies to your own wallet on your phone or computer.