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My understanding is that it's impossible for a spaceship or anything else with mass to actually reach the speed of light. It can only approach it. Only massless energetic waves like light and radiation can travel at the speed of light.
Your understanding is correct.
Relativistic mass increases the faster the moving object gets. That in turn means more energy is required to accelerate an object the closer it gets to the speed of light.
Fun fact: the speed of light is not as absolute as it might seem when looking at relativistic effects. In media with a refraction index above 1 (only perfect vacuum has a refractiom index of 1), the speed of light equals 1/(refraction index).
For light moving in water that results in a speed of light of around 3/4 the speed of light in vacuum.