If my ship gets destroyed, do I lose my skills or my character's progress?

No. Skills are never lost when you die.

• When your ship is destroyed, you lose the ship and whatever was fitted to it or in its cargo.
• If your pod (capsule) is destroyed too, you wake up in a new clone at your home station. Your skills are untouched; only implants plugged into that clone are lost.

That is why the usual advice is: fly what you can afford to lose. Losses in EVE are tuition, not a reset.