I'm having troubles starting my car.?

Without examining the car it's impossible to be precise, but if it can be push started as you state and the alternator and battery are good then the most likely culprit is the starter motor assembly which on modern cars also incorporates the starter solenoid. If there's just a click from the starter that's a classic sign of a failing solenoid. Usually a sharp tap with a hammer on the solenoid will free it off and allow starting; often best done as an assistant tries the key. The solenoid is the smaller parallel cylinder on the starter motor.

Check all connections to the starter and to the solenoid and also check the ignition switch is sending 12v to the solenoid when it's turned to the start position. If the switch is good then the starter motor is most likely faulty. If the switch fails to deliver 12v to the starter solenoid then either the switch itself is faulty or there's a loss of continuity in the wiring to the switch or from it to the solenoid.

Methodical diagnosis is vital.

2

first, your problem is in the ignition switch, maybe just a fuse, ck. ,Fuse panel. Second, your bat.t will accept a charge from the alternator of 14.0/5 when it is being used with a lot of accessory's being used. your alternator puts out a charge of 12.5 when there NO accessory's being used at curb idle.
My money is on your ignition switch. I would repl. that switch no matter what!!
john r

I'd rent a vehicle before taking a car that obviously is going to leave you stranded somewhere. Leave it at the dealer and let them drive it so they catch the problem and have it fixed for when you get back from your trip.

If it won't turn over but will start when you jump it, it's the battery!