You will want to know which jurisdiction kept the records you are trying to find so you know which level to begin your search on. You can find out where records were kept in Ancestry's Red Book: American State, County and Town Sources edited by Alice Eichholz, or in County Courthouse Book by Elizabeth Petty Bentley. Granted these are in the old-fashioned book form, but they have great information on what jurisdiction you should be consulting. If you prefer to use the Internet, you should be able to find information on jurisdictional holdings by using a search engine.
If you would like to just browse the site of a government entity to see what they have available, there are some general formats for government websites that will get you pretty much where you need to be without having to sift through the results of a search engine. When trying to find the site of a national institution, the you can generally use www.acronym.gov (example: www.nara.gov). When you are trying to locate a state government site use www.state.stateabbreviation.us (example: www.state.ca.us). For county governments go to www.co.countyname.stateabbreviation.us (example: www.co.greenlee.az.us). If you are unable to get to the page you would like to view, try www.statelocalgov.net and you can search by state for the official state, county, or city/town site.
When you are unable to find the records you need on these pages it wouldn't hurt to consult sites such as Cyndi's List or USGenWeb to see what information they can give regarding where records might be kept. Unfortunately, not all records are available online at this time, but there is a good portion of them, depending on the area you are searching. When they are not available to you from home, don't forget that these sites might also be able to tell you how you can get access to the records you need.
As you continue to research a family and are trying to use online databases, remember that information is usually added in chunks, not all at once, so it would be a good idea to check any site periodically to see what has been added—some sites will have a calendar of when they expect their information to be available for you to search. Another good thing to remember is that when you are searching on the state level, many times the Secretary of State's homepage will have the most records that are useful to family history research. You will also want to watch out for links to historical societies or libraries with special collections while you are on government pages as well. These may be just what you need to solve that really tough research problem.