In car neon light sticks

cut the plug off. There should be 2 wires exposed. One is hot one is ground. "People with mullets live 40% longer"   - Ricky Bobby

First, check the specs of the neon tubes to get the amperage. You will need an adequate switch that can handle the amperage of the 3 tubes(unless you hard wire them in parallel to an existing circuit). Cut the plug and enjoy. I prefer cathodes myself, less amperage draw than neons. http://www.oznium.com

I might suggest instead of NEON, you look for L.E.D. tubes. They will take much more of that accidental abuse, and a much easier install. No current draw.

Neons can take some time to light up, it's just the nature of the gass, but it could also be a problem of overloading a circuit.  Did you tie them into an existing circuit or did you run a new power wire?  If you tied them into the same line that your gauge cluster runs off then that could cause issues with the gauges as well.  It's always best to run a new power wire when installing aftermarket lighting, you save yourself a lot of issues down the road, it's easy to expand and you don't have to worry about frying your stock wiring.  In terms of amp draw, sets differ, you should be able to find out based on the packaging or from the manufacturer.  Draw is a big reason a lot of people go with LED or Cathodes.  I have four cathode bars in my car and they draw next to nothing.