This is where the name relativity comes in. You have to think in terms of relative speed. Your speed relative to earth will indeed advance closer and closer to c but never reach it. There’s a bunch of really wild and crazy implications behind this.

ah, right. In a ship travelling with c, for someone outside the ship, I turn on the lights and observe the light to travel with c. For that external observer the light from my lamp travels at the same speed as my ship

My mind was already bent! ;)