Photography is all about capturing light, so shooting Neon signs should be easy, right? Well, not so much. When the light source is the subject, they way your eye sees the subject, and the way your camera sees it are very different. The light meter in your camera is essentially looking for mid-tones. When what you're shooting is so high contrast with lots of shadows, most of the time, the camera will blow out the neon tubes and render the shadows a grainy dark gray. The following is my approach. YMMV, depending on the look you're going for.
Underexposing is what you want. It keeps the shadows nice and black, but holds the detail in the neon tubes. I stick with manual exposure and usually ignore the meter. My method started with a lot of trial and error, but after enough practice, I've narrowed my settings to an ISO of 400 or 800, a shutter speed range of 1/90 to 1/180, and an aperture of f4.5 to f8. I usually start with ISO 400 and 1/125 at f5.6, then adjust to taste.
For post-processing, My goal is to hold the highlights of the neon tubes, and correctly represent the colors as I saw them. First off, I always shoot in Raw format, and process using the Photoshop Camera Raw module. If it's true neon (orange-red with yellow highlights), my goal is to hold the yellow highlights, and the detail of the surface directly behind the tubes. Adjustments to contrast, clarity, and color hue adjustments to reds and oranges, but go easy with the hues, or things get out of whack quickly. Lowering the highlights a bit can also help. These adjustments are to taste and can vary depending on the shot, so plan on doing a bit of tinkering, and save those settings once you get the results you want.