The best thing to do is to come up with a domain name which is not currently registered. To find out whether a domain name is currently registered you can search using sites such as https://www.whois.net/

For a UK .co.uk search, use the facilities provided by nominet at https://www.nic.uk

The reason that this is the best option is that once a domain name is already registered, it is always more expensive to buy it, if in fact the owner will sell it at all.

You should also think of the different extensions that you might want to register. For example US companies wanting to also sell in the UK would want to register the .com version of the address as well as the .co.uk

Buying a name which is not currently registered

You can register domain names with many domain name companies. In the USA you might use https://www.godaddy.com

in the UK you might use https://www.123-reg.co.uk

Sign up to an account with those companies and register the domain name. They will usually try and sell you lots of other services when you register the name. The only other service you might consider using from those companies, is their email services, as Sellr does not provide email accounts.

Buying a name which is already registered

Firstly navigate to the domain name using your browser. Many domains which are not in use will have a holding page with details on how you can buy the domain name.

If there is no holding page then you will need to find out who owns the domain name to try and contact them directly. Use https://www.whois.net/ and https://www.nic.uk to view the details of the current owner.

If their details are masked then you may want to try emailing info@thedomainname.

Buying a name from a domain marketplace or domain broker

One of the most popular domain marketplaces is https://www.sedo.com this is a great service where you can view thousands of domain names and purchase them. Sedo will also manage the financial side of the transaction managing the transfer, so it is less risky than buying from a private individual. In fact, if you do find a private individual who will sell you a name, you may still want to manage the sale through a site like Sedo.

There are also domain brokers who have domain names which you could use to build a brand. One such site is https://www.brandbucket.com which is worth looking at to get some ideas.