LocalBitcoins is hands down our favourite option. Handling millions of dollars worth of transactions each day, it's the main meeting place for everyone looking to either buy or sell Bitcoin.
If you find it difficult to use, keep scrolling.
AnyCoinDirect is a service based in the Netherlands that sells Bitcoin and offers a vast range of EU payment methods, including Bancontact/Mistercash, EPS, Giropay, iDEAL, Mybank, SEPA, Sofort and TrustPay.
Coinbase is a trustworthy service that sells Bitcoin and accepts most credit & debit cards. They do require photos of your ID for Know Your Customer (KYC) and Anti Money Laundering (AML) reasons, but they are one of the most versatile services out there.
Another EU alternative would be Indacoin (much pricier, but they don't require registration and don't do tedious KYC & AML checks).