There are thousands of companies online that offer hosting plans and domain registrations of varying quality, for varying prices. Over the years, I have used many different companies (some commercial and some free) and I have had varying degrees of success with them.
A good web host can offer many features (the main ones are the amount of space provided and the scripting languages supported) and be staked full of options (such as adding domains and content management systems), but a great web host, most importantly, offers good support. Good support is provided when the response is rapid (a full 24 hours at the most) and contains the correct information that is useful and relevant to the question.
Choosing the correct domain name for your website is crucial, as it needs to be related to the subject, easy to remember and it must send out the right message. Although, of course, with Wattersisere.co.uk I am a hypocrite. The company you choose is very important. Firstly they must offer the suffix you want (the country code, for example .co.uk) as it not only tells the customer about location, but about business type (.org for organisation). Secondly, the features must be correct for what you want, and once again the support provided is critical.
I advise using the following companies, though there are many more out there, some of which I have not come across yet: