1) Lighting in a home is never wired in series, they are always wired in a parallel fashion. Daisy chaining from 1 light hot and neutral to the next downstream light hot and neutral means they are wired in parallel.
2) The power source only lands on 1 black screw at one of the 3-ways and never to the 2 brass screws. Shown in the diagram I provided is exactly how two 3-ways need to be wired.
3) More than likely, you have the two traveler wires crossed up at the black screws or spliced to some other wires.
4) If your meter has a continuity feature, you need to confirm the two travelers landing at each brass screw are the same 2 wires. The breaker must be shut OFF in order to check for continuity.
5) Do you have a constant 120 volts at the black screw on one 3-way whether either switch is ON or OFF?
6) Power needs to originate at only one 3-way. Power can originate at a light box, but then must be extended to only one 3-way. Power cannot originate at a 3-way and simultaneously at a light. The light is switched by either 3-way.
7) Do the 2 traveler wires originate from the same cable sheath?