I have been using Hostgator for wordpress without any issues. Dotster.com are also good for hosting and you can save 15% with coupon "bargaincat.com". Another host is Lunarpages that offer wordpress installs.
LCN have a wordpress blog hosting package: http://www.lcn.com/blog_hosting £20 per year. Not sure if that fits into your budget but it's got all you should need within one pacakage.
since you haven`t specify ... why don`t try Blogspot.com or wordpress.com it is free and you cam set up a blog in just about 5 minute with no knowledge at all.
Depending upon the nature of website(Wordpress I mean) and the expetations out of it in long run you must prefer established hosting like "Godaddy","Hostmonster",Hostgator" and others even if you have to shell out a little more. If you are simply a blogger who is writing for self pleasure and nor primarily for business objectives then you may shift your focus to hundreds of cheap hosting, share hosting,or reseller hosting offers.But do not forget to check the authenticity of the offer first.