Electrical problem/no start

2003 Taurus, approx 86,000 miles, a/t - We've had no problems on this vehicle til this week. All fluids were topped off, battery terminals cleaned and battery water added on Sunday. Car drove fine Mon thru Wed and started okay Thurs Am. When left not running for a few minutes then tried to restart, nothing happened. Various clicks from engine while turning the key and from different locations and sounding different every time. Also different behaviour from dash, interior and outside lights. Car would not accept a jump. Towed home and replaced battery. Same problems. When working the automatic locks/outside lights via the key fob, get various reactions such as some work, all work, none work and varied strengths each try. When key turned to start position, various reactions of dash lights/idiot lights also. Car does not always click, and still does not start. Can you narrow down what I need to check before I start taking things apart, please?