Toyota Repair: electrical problems, brake light switch, toyota corolla

Question
We have a 1997 toyota corolla.  It has been doing some strange things...the first thing is that when I stepped on the brake pedal, the tape player light and the dash board lights (the speedometer, odometer, etc.) will come on and the clock will dim.  This will happen whether the car is started or has a key in the ignition or not.

And also, when the car is started, the brake lights are on continuously.  When I hit the brake pedal, the brake lights brighten but do not ever go off. (Unless the car is turned off.)

We did have to replace a battery a while back but the battery was old...just thought it was the time for it.  The problem started apx two weeks later.

Is there any place specific to look for the problem?  Thank you for your input.  

Answer
Were any bulbs replaced recently or was there anything at all done to the car after the battery replacement?
If not and this problem started by itself try replacing the brake light switch, it may have developed an internal problem and is creating feedback voltage through the other circuits.