You can get the forms from Companies House and do it yourself paying the relevant fee, which is the cheapest option (I think all the forms are available online to download), or you can buy a so-called 'shelf' company from a company registration agent who will register a change of name to the name you want and register the change the shareholders and directors for you (costs around £100), or you can get a company registration agent to register a bespoke company for you (most expensive option).
You can find company registration agents by doing a google search.
Companies House fees are £20 efaze is £30, you do it online. I don't know who they are but have been using them at work (am an accountant and incorporate new companies for clients - we use the agent to save us filling everything in and it gives you all the paperwork) for at least a couple of years.
the main benefit, as the name suggests, is that it is limited liability - ie if it all goes pear shaped then the creditors can'r (in most cases) pursue you. This isn't as good as it sounds as any creditor with any sense (banks etc) will insist on personal guarantees if the company doesn't have it's own assets. Other potential benefit is tax, depends on the profits. You should talk to an accountant, they will give you set up advice - which is the best way, and can advise on what records to keep to make end of year accounts easier. Plus other things.