Check your battery cables. Were they really filthy? Make sure ground is tight and ALL wires are on or attached to the solenoid. Check the hot wire going to the inside of the car (fuse block, wire to the ignition and wire to light switch are hot). The positive cable from the solenoid is connected solidly to the starter.
The alternator doesn't enter into the equation until the engine is running; but it does need a hot wire going to it to.
my first suspect would be the altinator. replace it. its an easy job. if the altinator is dead it will not charge the battery. no charge in the battery then the lights will not work. the whole car will be dead. this is exactly what you said you were experiencing. jump the car. if it takes off then replace the altinator it is dead. if it takes off then all of the connections to the celenoid are good and the starter is functioning properly. rhe altinator does not become the main concern once the car is going it is the main concern because the car is not going.