Here are where many start on their journey in pursuit of a name.
There are a couple of large databases on the net that are run by "pay sites". This does not mean you will ever need to pay them a dime. The initial search is free and this info can lead you to other free avenue's.
The first is

Ancestry.com - This is a pay site, but the search is free. It claims to have over 550 million names in its database. This search, if successful, will provide links to free areas of the site, and also areas that are paid. I strongly discourage you from paying for anything at this point. To much is free to be paying for information. At a later time, you may find you really want to buy something.

Search the Ancestry World Tree
Surname
Given Name

Family Tree Maker - Another pay site. But again, the search is free, and the hits it comes back, with may be free or pay.  

Root's Web - totally 100% free. A great organization and as you get more and more into genealogy, you will find you are coming across more and more of their sites.
These are the best that I have found, but I am open to further suggestion.

Surname
Given Name


Probably the most success you will have in your search, is finding others working on the same name you are... or maybe they have already traced further back. Most people are more than willing to share what they know. The forum's have been my best single source of information so far. There are forum's for every name, every state, and every county. You just have to find the right one :-)  Here is where to start.

Looking for a name -

Surnames (the last name) - 100% free! Here is where you can find links to the first letter of the name you are looking for. Or better yet, in the forum finder box, just type the last name, or state, or county you are looking for. When you find the forum you are looking for, type in the name you are searching for. Read what others have written about people with this name., You may find a match, or a clue. When you have searched the entire forum, leave your own enquiry totally free. Who knows... a cousin you never knew you had may write you. And don't be impatient here... it may take months for someone to come across your enquiry and write back. Or days. You never know.

Surnames - part of a pay site, but still you can use the forum for free. Kind of temperamental and slow..but worth a shot.

AOL also has a small but sometimes useful forum for its members. Just type "Genealogy" under Keyword.

No, we aren't don't yet. Much more to check on page 4.

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