'Family Search' is free and owned by the Church of Jesus Christ of Latter-Day Saints. I use it a lot and it's a good place to start I think. You can begin to build up your tree there.
I subscribe to Find my Past and Ancestry.com where my main tree is stored with photographs and other documentation. It works out quite expensive, but it's my hobby and I enjoy exploring the lives of my ancestors. You will be able to find a certain amount of information and then you'll reach a file with a lock! In order to probe deeper, you'll have to pay. One night after weeks of searching and turning back when I reached the lock symbol, I reached for my credit card and there they all were - my family.
There is an Ancestry.co.uk but I chose .com because most of my ancestors, with the exception of my great grandfather, emigrated to the US/Australia.
Tonight I'm working with all three open and cross-referencing.
A good place to start is Cyndi's List where a very wide range of sites are listed.
Internet is an amazing tool for finding who you really are. Top tip: if you have someone with an unusual name follow them!
I hope you enjoy finding out about your family and wish you every success. I found many cousins we didn't know we had and now family on both sides of the Atlantic are in touch and meet up whenever we can. Good luck!