2001 Honda Civic difficulty starting when cold

Hello, my 2001 honda civic has had starting problems that seem to be worse depending on how cold it is outside along with how long it has been sitting. Basically I have to turn the key and hold it for as long as 10 seconds before the engine turns over - no clicking or revving, silence until it finally turns over. It starts up right when the alternator turns over and it never stalls while running. I know its temperature related because it sat idle for 10 days but started right up when I got home and started it in the afternoon (warm in Arizona). My battery is new (replaced it about 3 months ago). My coworker and I tried removing and cleaning off any corrosion from the ignition switch, but this did not work (and we didnt notice any corrosion). It seems to me that it has to be an electrical problem where the voltage is not reaching the starter until the key has been held for several seconds and the power builds up and surges through a bad connection. When the key is on the "on" position the lights on the dash light up so there is some power although the engine won't start. I would love your advice on what to check next/replace so that I can do it myself if it doesn't require a lot of dis assembly. Thanks very much.