For all adoptees and others with unknown parents and/or grandparents, I recommend autosomal DNA testing.
Males with unknown paternity should also order a Y-STR test (Y-37 level or higher). With this test it may be possible for a male to determine his genetic surname. Max Blankfeld of Family Tree DNA has estimated that this is possible about 30%-40% of the time.
For autosomal DNA testing, start with either AncestryDNA ($99), 23andMe ($99) and/or
Family Tree DNA's Family Finder ($79). Recently, we have been seeing a lot of success with AncestryDNA and unknown parentage cases due to the large database and connected family trees. If at all possible, test at all three companies in order to "fish in all three ponds" for close family matches. There are so many success stories using this method that it is important to give yourself the best chance possible because you never know in which database the key match will be found that will solve the mystery of your origins! AncestryDNA data may also be uploaded to FTDNA's Family Finder database for free (you get all of your matches and it is only $19 additional to receive the enhanced features, like the chromosome browser, My Origins, etc...), so if you test there first, you can be in both of those databases for much less.
Find 23andMe International here.
AncestryDNA's strength lies in its vast collection of family trees that are attached to your matches' DNA results. I strongly recommend their product for unknown parentage/adoption search.
For Y-STR and mtDNA testing, Family Tree DNA (aka FTDNA) is the only company that I can currently recommend. I generally recommend testing, at least, 37 markers for Y-DNA. If finances are an issue, then start with one of the lower resolution tests and upgrade at a later time.
I generally do not recommend mtDNA* testing for adoptees except in rare circumstances. If you decide to use a mtDNA test in your research, then I recommend the full sequence mtDNA product.
*Mitochondrial DNA (mtDNA) testing is usually not genealogically informative, so I recommend that women, instead, start with autosomal DNA testing and men test either their Y-DNA or autosomal DNA (or both).
For researching DNA matches' family trees, I find Ancestry.com to be invaluable:
Spokeo is very helpful for locating living people once you have narrowed down your search. Click here >> Spokeo
US Search is also helpful for researching living people.
Newspapers.com is great for searching for obituaries and finding marriages, which are essential when researching living people. I use this service every day.