Old situation:
- Power entered an old light box via 14/2.
- There was a 14/3 from that box to the switch.
- The black and white to the switch supply outlets downstream of the switch.
- The switch was connected between the black and red.
- The old light was connected to the red from the 14/3 and the combined white. The switched power to the light traveled 'backward' along the red to the light.

New situation:
- The old light has been replaced in a remodel.
- There's a 14/2 from the new first light to a second light.
- MsGreen1 has extended the 14/3 to the new first light box with 14/2, disconnected the 14/3 black from the upstream power and connected it to the 14/2 extension black that is connected only to the lights.
- The outlets downstream of the switch are now fed by power *through* the lights when the switch is off, and so they never work right. When the switch is on, they're shorted entirely out of the circuit.
- Guessing: The switch is now probably connected on the neutral side of the lights, which is just plain wrong, and potentially dangerous when changing bulbs.

The only solution is that you need to remove that 14/2 extension of the 14/3 and replace it so that you have 14/3 all the way from the first light to the switch. Connect it so that:
- All whites are connected together in each box
- The 14/3 black is connected to the power source black
- The 14/3 red in the box of the first light is connected to the light fixture black
- The 14/3 red is also connected to the black wire of the 14/2 going to the second light.
- The switch box is reconnected the way it was, with white-to-white, black-to-black, and the switch between blacks and the 14/3 red.

I don't have time to draw right now, but here's a picture from someone else. You'll have to imagine connecting the cable to the second light in the same places that the light shown is connected.