This is how it works in Georgia. Could be slightly different in your state.
You need to go to your county court. They will have about 3-4 forms to fill. They are usually very simple forms, where you mention your current name, new name you want, , address, reason for name change etc. Then give those forms to court. The take fee (78$ in Atlanta) You have to pay for an ad in news paper for 4 weeks. In GA, when you file the case, they take extra $80 from you and send the ad themselves to correct paper. At the same time, clark will assign you to next judge.
After 4 weeks of ad is over, judge's office will schedule the hearing on your name change case. They usually try to club all name change cases together and may have only once a month or so (Depends on judge) On the day of hearing, you go there, go take the stand when your case is called. Judge will ask some questions (Why name change etc) and grant the name change by signing name change order. You can go to judge's clark and get more copies of that order.
Overall process can take 2-3 months (1 month for ad + some more weeks to get hearing scheduled) May be faster too for some judges and your timing of filing.
Hope that helps. If you have any specific questions, post them here