I'm also from the EU and used to have BTC (now I'm into LTC), so here is how I do it:

Register in Bitstamp or Kraken (my favourite exchanges).

Validate your account. You have to provide some sort of proof of identity and address. This is asked by governments and if they don't comply with them they can be closed without any previous notice, like it happened with the exchange bitcoin24. It can take up to 1 week.

Deposit cash. Companies like Visa/Mastercard/PayPal don't want to deal directly with BTC, that's why a bank transfer is more convenient. Being European you probably have cash in a SEPA bank, which is an advantadge because most exchanges accept SEPA deposits at a very low fee (much cheaper than a Wire Transfer) and they are usually very fast (1-2 working days).