Researching your family history is now much simpler because so much information is available on the Web. Sources range from searchable transcripts of entire documents to general study aids and information about archive collections held in repositories. Much of the web content has been created by archives services and is provided free; some sites make a charge for specific services; some commercial services charge for access to some or all of the Web information they have created.
The best advice to those starting out in family history is still: read a book! There are also a range of materials on the Web: the first point of call should be the family history pages of The National Archives, Gathering the Jewels or the BBC website.
Family historians have focused on registers of births, marriages and deaths, wills, and census records, since these were the best way to locate individuals. Indexes to many of these records are now available on the Web.
Resources like Archives Network Wales (this site) provide a different sort of service, covering the whole range of archival materials (business records, estate papers, personal correspondence, charity and local government records, oral history recordings, maps and photographs). Because only a summary of the detailed catalogue is presented on the website, this type of resource will not usually yield results if searched for an individual name. For example, a miner's name will appear in wage lists and other mining company records, but searching a collection-level website such as ANW for the name will draw a blank. However, a search for mining in an area as a topic, or for the relevant parish or company name, will allow you to locate the repository holding the records. Help on searching ANW.
Only a tiny fraction of archives in the UK have been mounted on the Web. But information about archive collections can now be searched and examined much more easily, which should mean that your visits are much more efficient and enjoyable. There are resources like ANW covering much of the UK.
More links are listed on the New to archives page.