Looking into buying a house?

So I am really wanting to get my own place for me and my daughter. I don't really know where to start. When you buy a home can you buy one without a mortgage? Im going to get a second job and work as much as I can to save as much as I can. Looking around at houses for sell I could have enough to fully buy a 20,000 home by end of next year. Now if I were to pay full for a house that wouldnt give me monthly payments right? I would just have to worry about electric and water ect?