There are several different genealogy libraries and centers that you can go to in order to complete you research. One of the advantages of using a center to do your research is that it is staffed by people who can help you when you run into a problem in your research. If you are just beginning going to a center can help you begin to get started. Many of the centers offer classes that can help you get started.
Another reason to go to a center or library is that you can order microfilm or microfiche through them. They will also have the equipment that allows you to do the research. Additionally they often have subscriptions to databases that they allow the public access to when at the library. This means that you can expand your research by visiting a library.
In many state and local library systems there is often one library that has a genealogy center located in it. This is usually one room with several computers and a staff that has been trained to help you with your research. The best way to locate one is to contact your local library system and check to see if they have this available. Often these types of libraries focus on the records for that particular region, though they can help you order material for different regions as well.
Genealogical societies in your area may offer a small library as well. Although these societies often work in conjunction with local libraries to keep the genealogical library open, so they may not have their own library. However it is worth looking into it in order to find a research.
Another resource are the Family History Centers offered by the Church of Jesus Christ of Latter-day Saints. These are run by volunteers and may have a unique set of hours, depending on how busy they are and how well they are staffed. Although a church runs these centers, there is no proselyting or discussion of religion at the centers, so you should not let that discourage you from utilizing one. They are there for the public to use as well as church members.