/// This sounds like a ground problem as one of the other posts suggested. This can be a bad connection at any joint in the primary, battery charging circuit. Since the lights are dim at first switch on, they should all light before you crank, it is probably there.

With an assistant watching the lights use a heavy wire to jump the cables one at the time. From battery negative post to engine ground, would be my first choice. Then bypass the system wire from the positive post and follow by testing the connections and wires to the alternator. You are looking for a bad connection inside the lead/brass battery post clamp. You cannot see the condition of that connection.

You can measure the voltage drop while the car is cranked and then while running. Any wire that shows a drop of more than one or two tenths of a volt is suspect.

Bear in mind that your meter may be off, test it. A battery reading in the morning after overnight rest should be 12.66 volts, 12 volts is half charged. If you have charged the battery burn off surface charge by using the headlights for a few minutes, then read voltage.

The system must be loaded to force the alternator to charge especially with a new battery. After cranking the charge time may be very short. Turn on lights, radio, AC to force charging voltage.