I would see if your local college/university/community college is offering any classes in GIS. A lot of them do nowadays and many have majors or its now part of their Geography major. (As in my case) Also, ESRI courses are also a good place to start. You can take them online or see if there's someone in your area teaching them in person. Check out their courses here: http://training.esri.com/gateway/index.cfm
Just being proficient in ArcMap/ArcCatalog and so on is really good, but if you want to get into the programing side with VBA or python people are always looking for those kinds of skills.