Note that each country in Europe has their own cell phone plans. Many companies, like 3 or Vodafone, operate in more than one country, but they are still different plans. So, Vodafone in the UK will have different offerings than Vodafone in Spain, and you probably can't get service for a UK Vodafone account in a Spanish Vodafone store (any more than a T-Mobile store in Germany will be able to do anything with a T-Mobile US account).
However, within the EU, there is no longer any extra charge for roaming. Meaning, a SIM bought in London (UK) will not cost any extra to use in France, Belgium, the Netherlands, or Germany. You do have to make sure that the account on the SIM has roaming enabled (it's not always set by default).
You also have to check the terms of the EXACT plan you are getting. How much does it include of texts, data, and calls? Where are the calls to? (some plans have cheap calls to other countries in the EU, but expensive calls to non EU countries such as Switzerland or the US).
You can buy SIM cards at the stores of the various carriers (EE, Vodafone, etc) or can go to a Carphone Warehouse store, which handles SIM cards from multiple carriers. Here's a page from Carphone Warehouse detailing some of their prepaid SIM options; you'll see that for £10-15 you can get a lot of data, along with varying amounts of calls and texts: https://www.carphonewarehouse.com/sim-only/pay-as-you-go-sim.html