Photographing through glass is one of the harder skills for a new photographer to learn. Not only must the photographer take care to avoid his own reflection, he must also strategically position his light sources so light is not reflected in the glass. Additionally, because glass is prone to showing fingerprints or smudges, the photographer must be aware of any of these types of distractions that may draw the viewer away from the subject of the photograph.
Wipe the glass in front of the painting with a microfiber cloth to remove any dust, fingerprints and smudges that may show up in your photograph. If this is not possible--for example, if the paintings are in a museum that will not permit you to touch the painting--you may have to digitally remove any smudges that appear in the final photograph.
Attach a polarizing filter to the front of your lens. A polarizing filter will filter light perpendicular to the filter and remove reflections that may have otherwise been visible in the glass. Look through your camera's viewfinder and rotate the polarizing filter until the reflections disappear.
Use an external flash to illuminate the painting. On-camera flashes will bounce back at the camera when you take the picture and will appear in the photograph. Hold an external flash unit at an angle to the painting so that the reflected flash bounces away from the camera's lens. Imagine that the flash is a laser pointer aimed at a mirror, and direct the flash so the "laser" will not hit your camera. If you do not have an external flash and must use an on-camera flash, position yourself and your camera at an angle to the painting, usually about 45 degrees.
Verify the focus of your photograph by using the live preview screen on the back of your camera. Some autofocus mechanisms will be confused by reflections in the glass, and will focus incorrectly on the glass itself or on the reflections. Make sure the painting itself is in sharp focus before moving on.
Move either as close as you can or as far away as you can from the painting to minimize the effect of your own reflection. Your choice will depend on the particular venue (how close you are permitted to get to the art work, and how many other patrons would be in your way if you were to move away from the painting) and the lenses that you have available (either wide angle or telephoto lenses).