It goes without saying that you should first study art. You should know all about the different artists and their work, including the different time periods throughout history. Having a strong foundation of art will definitely help you find a job in a museum or art gallery.
Look for an internship with a museum or art gallery in your area. If you cannot find any type of internship then look for something that has something to do with art. Working in any organisation that deals in art will definitely help you get some much needed experience. Be sure to work hard as they might offer you a permanent position after your internship is over.
Make a comprehensive resume detailing your educational and work experience. If you are a fresh graduate then you can always put down any internship that you may have completed while studying art. Your resume should be concise and easy to read.
Go online and search for different museums and art galleries that are located in and around the area you live. If you are okay with commuting then also search for museums and art galleries in nearby cities as well. While you are online, visit the different museum or art gallery websites for job opportunities. Apply to as many as you can with your resume. Also, look for job placement websites that cater to art galleries and museum employment opportunities. Make an account on these websites and apply to the different jobs listed.
Check newspaper and trade magazines:
Go through different newspapers and trade magazines for possible job listings. You will find some employment listings here so be sure to apply to as many as you can find.
Try to network with as many different art gallery employees and owners as you can. Visit different galleries and art shows to build up your contact base. You will find out about any opportunities that might exist.
Finding job always takes time so it is a good idea to be patient. If you have applied to different museums and art galleries you will hopefully get a response soon.