I’ll start by assuming that you are using a simple digital camera. If you are using film, a medium format or view camera I think it’s a safe assumption that you know what you are doing. For the rest of us it’s all about light. Your camera will do the heavy lifting but there are a few things you need to control.
Find a location where there is enough light that your cameras flash does not go off and the shutter speed is high enough that the images are not blurry. You may need to turn the flash off. Flash will create a lot of unwanted reflections. Put the painting in front of a window so the window light is falling on it. North light is best, not direct sunlight. You want diffuse light so if the window has white sheers that is perfect. If there are no sheers then try to have the light fall on the painting from an angle right and/or left. The idea here is to have no shadows on the painting and to have no shadows from the paint texture.
Use a tripod if you have one. If not it’s important that the room be bright so that the camera will use a high shutter speed. Try to have the camera lens parallel to the painting so one side of the painting does not appear larger than the other. Use a little zoom. Most cameras do not have rectilinear lenses so there is often a little barrel distortion. The outer edges will appear curved. Now take a few pictures. Take a lot, it doesn’t cost anything.
If you have access to Photoshop you can take several images like a mosaic and Photoshop will put them together as one large higher-resolution image. There are other applications that may be able to do this, even the application that came with your camera. You may not need a high-resolution image depending on what you want to do with it. If you want to use it to sell hi-quality prints then it might be worth having a professional take a good high-resolution image. If you just want to put it on the Web any digital camera and many phones should be fine. All you need for a good Web image is about 72 pixels per inch of resolution. So if the image is going to be 10 inches by 10 inches on the screen then the image needs to be 720 pixels by 720 pixels. Most cameras are at least this good and if your camera is better than that most Websites will accept higher resolution images so you shouldn’t have to worry about it.
Now that you have pictures you can use them as is, or do a little adjustment. Try to get the images to look as much like the original as possible. Since you cannot get the image on screen to match the original exactly, good enough is the goal. The image can also be cropped. I use the automatic colour, tone and contrast settings as much as possible. Keep an eye on the image and undo if you don’t like the result.
Now you have a perfectly usable image of your art.
Take a class or buy a book. There are good ones out there.