As you seem to realise, you'll need to register a domain name (= the bit after the @ sign) and you'll also have to get someone to host your mail server. You never really 'buy' a domain name, since the registration has to be periodically renewed (so you're effectively 'renting' it).
Domain names ending in 'uk' are controlled by Nominet but (unless you have specialist technical knowledge) you have to register via a registrar:
http://www.nominet.org.uk/registrants/
Since Nominet charges a (small) fee for registration, this is nearly always passed on by the registrar. The only exception will be when you sign up for a package which includes the fee.
There are a few 'free' email services, which include your choice of domain name, but they can be unreliable and also tend to pay for their services by appending advertising material to your mail. (That doesn't look very professional when used for business purposes).
Your best option is to sign up with one of the 'big boys' who'll provide exactly what you want. The UK market leader (which is used by many major companies, as well as private individuals) is 1&1. They'll provide you with your choice of a .co.uk domain name for £2.49 (+VAT) for the first year, and £2.99 (+VAT) thereafter. They'll also host your email for just 69p (+VAT) per month:
http://tinyurl.com/yfn92rv
Chris