EDIT: LOL, American. In that case you won't be able to use some of the websites, but try www.genesreunited.com (not .co.uk), and try Googling the rest. I'm sure the principles are the same in the States as they are here in the UK.
I know you said you wanted free searches, but you will probably find you have to pay for some things. You have to be prepared to put some money into searching for your sibling. If it will lead you to him/her, then surely it's worth it?
www.genesreunited.co.uk shows all births registered in the UK up until 2005, but you have to be prepared to pay to search through the records. The costs aren't too bad. You can buy "credits" for £5, and they expire after 5 days. Each record you view costs either 1 or 5 credits (I don't remember). But if you know roughly when they were born, and obviously their name, then you won't spend too much in the searches.
They organise the results alphabetically, then by year, then by quarter (eg Jan-Mar). I assume the child would have had your surname, and it should also state the mother's maiden name, and area of birth (Eg NW Surrey).
Other sites, like www.freebmd.org.uk show birth records too, but not as recent as you need I don't think.
Once you have found the record you need, make a note of the page references (eg 8e 406) and you can go online to order the certificate, at www.gro.gov.uk. Each certificate costs £7.00, and normally takes around 4 days to be posted.
I'm not sure about adoption records, as I've never had to research this, but after asking Google, I have found the following site: http://www.govtregistry.com/uk.php?sk=Adoption&tc=ml25l101
Once you find out the family they were adopted into, you can search through the electoral rolls to see where they were living over the years. This too does cost, but you might be lucky enough to find someone who already has a membership to search. Also you could try phone books.
If they're of marrying age, you can search for the marriage records (again a good place for this is Genes Reunited). If your sibling is female, you would then find out her married surname, and then trace her in this way.
Good luck, I hope you find them.