Preparation

Before you start investing, consider the following points. They will give you an idea of what your preferences might be and which website or app is the best for you to choose from.
After answering the questions, you can go here and find additional information for the most suitable option to buy cryptocurrencies.

  1. I am ok with sharing personal information (ID, Photo with Passport, Bank account, Phone number).
    • Yes: coinbase.com, cointed.com, bitpanda.com, bitfinex.com, cex.io, kraken.com
    • No:  cointal.com, localbitcoin.com, coinmama.com, paxful.com
       
  2. I want the transaction to happen as fast as possible.
    • Yes: All providers that allow to buy cryptocurrencies through credit card, paypal, SOFORT transaction
      (higher fees, lower limits): coinbase.com, cointed.com
    • No: Bank transactions (SEPA, SWIFT): coinbase.com, 
      (lower fees, higher limits)
       
  3. I want it to be as cheap as possible.
    SEPA and SWIFT transactions and websites with their own trading platforms (gdax.com, cointed.com)
     
  4. I want it as easy as possible and with a good user interface.
    Coinbase is by far the best way to go for beginners.
     
  5. Residence
    Depending where you are from, your options might be limited. Most options are currently available for European, U.S. and Australian citizens. 
     
  6. Security
    If you want to have a more secure way to buy cryptocurrencies, decentralized exchanges are the best. Check out robinhood.com (this only works for U.S. residents as of right now)

If you want to do more research, this website gives you a quick overview that helps you to make the best choice.
Here you find a website that also lists a ton of different platforms, but with the addition that it sorts them depending on your personal needs.
If you want to compare certain purchasing platforms this website might suit you. It lists exchange fees, traded cryptocurrencies, deposit and withdrawal options, security features and much more.