How about the grounds?
Bright interior lights is a good sign, but doesn't necessarily rule out a bad connection somewhere. Starting uses a heck of a lot more current than the lights.
There'* a few easy things to check. First, check the voltage at the terminal on the starter coming from the battery. It'* the largest connector and should read the same voltage as your battery. If you have another person to help, being VERY careful, check the voltage on the solenoid while someone turns the key. This wire here is enabled by the security system and should go to 12 V if your ignition is passing a start signal.