How to buy UK/EU stocks?

How do you buy stocks in UK/EU companies.  Is it as simple as getting something equivalent to an E*Trade account or do those not exist in the same manner?  Would you have a separate account/broker for each exchange or does anyone allow you to own stocks in multiple exchanges through the same account?  Apologies if this has been discussed recently and many thanks for any help.

I use idealing (www.idealing.com). If you're a British tax payer, you can get an ISA share account which means you can invest 7000 pounds and get any profits tax free.

You can trade on 19 different exchanges.

There are loads though. A quick google will find LOTS.