Where To Go

First thing you should do is locate the closest auto parts store to where you're located. For me that was O'Rielly Auto Parts. But, it could be any other store. Always remember that the internet is usually at your disposal and Google is your friend. Stores to look for could include any of the following stores, plus more:

Advance Auto Parts
Auto Zone
NAPA
Pep Boys
Car Quest

Any of these stores would be useful places for parts, accessories, and knowledge on how to do a repair. I also recommend that you travel to each of these locations near you and ask them what service shop they recommend for what car you have. Typically they will have more insight on who is trustworthy and does good work. Don't be fooled by mechanics, they're businessman and are often more concerned with money than the well being of customers. Keep those shops in mind when you encounter a problem you can't handle.

Now, if there is no rush in getting the part, you can often get it much cheaper online. You just need make sure you get the correct part. I typically use RockAuto Parts, eBay, or Amazon to get my parts. But there are many others out there.

Also, if you're looking to save a little bit MORE money and don't mind second hand parts you can go to your local junk/salvage yard and either buy it from them or pull it yourself. Depends on the yard. The only chain junk yard I know of is Pull-A-Part and they have constantly rotating inventory and good prices.