To start out searching for family history, write down everything you know, no matter how trivial it may seem.Talk to relatives and other people who knew the person you are looking for.Write down information such as parents, spouses, children, occupation, hobbies, etc.All those things may be a clue in the future.Keep good organized notes (you will eventually have your own system going).
Below are a few sites you might start with, but there are many more.
Find-A-Grave - http://www.findagrave.comhttp://www.findagrave.com Millions of cemetery records.Just type in a name, but you should know enough about the person to know if the right person comes up (time frame of when/where born/died, etc.).It has discussion forums.Sometimes someone can help you if you have a specific question and give enough information.It is not intended to be used as a means of getting someone else to do the research for you, but to ask "how to" or "where do I" type questions.
SSDI (Social Security Death Index) - http://ssdi.rootsweb.ancestry.comhttp://ssdi.rootsweb.ancestry.com Will give birth and death dates and, most of the time, where died and last residence.
FamilySearch.org - https://familysearch.org Lots of different records, such and census records (but not all).May find birth, marriage, death records, etc.Again, knowing the place and time frame would be helpful.
State historical society - some have on-line birth and death records, etc.Check for the state you're interested in.
Local historical society - May have obituary, if died locally, and possibly other information.Call, write, or e-mail them (check the internet for historical societies in appropriate location).
Local newspaper - Obituary, birth, or other news item.Small town newspapers are excellent for coverage on local people, that's who they generally write about (newspapers of 5,000 circulation or less).