I have been dabbling in researching my family tree for a few years now - I first started after seeing "
Who Do You Think You Are?". My starting point was simply to talk to family members - the older ones especially - so they can fill you in on the past couple of generations and hopefully the current ones, if you don't know your family well.
Collecting any family-related documents that may be around certainly helps too - birth, marriage & death (BMD) certificates help clear up a lot of mysteries or even start some new ones. Obtaining certified copies of these doesn't cost a lot (currently £7 per copy in England) and can help extend your search back greatly.
Recording your findings in either a dedicated family tree program or on one of the many websites is pretty much essential - though keeping a paper copy for annotation & as a working guide helps get the information laid out and is easy to quickly check & correct. Sharing your tree on such websites can reap benefits too - I have established contact with many distant relatives through doing so. However, ensure you set the privacy level so that you don't give out details of living relatives - that is often a serious concern if people suddenly find a Google search of their name produces a lot of personal information that shouldn't otherwise be available. Although some sites do charge for memberships, they do provide access to valuable information - transcribed census returns & BMD index information, military lists, maritime records and emigration records etc.
As well as simply recording the bare facts, finding and digitising old photographs helps flesh-out the members of your family tree, as does recording anecdotes or facts about the individuals as they come up during your research. There are also amateur enthusiasts who have devoted a lot of time to various research activities - one near me has photographed and transcribed all of the Memorial Inscription (grave) details for an entire cemetary and uploaded a searchable index online. Make use of whatever resources you can, but of course, be sure to thank those who help you on your way.
The key thing is to pace yourself and to keep enjoying it - finding a record after weeks of searching is a great feeling.