The photons we observed from the death of the star experienced no time between coming from the star and arriving to us.
Obviously, given that, you would have no way of stopping, so your arrival would be abrupt (ignoring whatever magical affects would be necessary to accelerate a massive object to light speed).
If you were able to do so, yes you could "get" anywhere in zero subjective time - as mentioned earlier, since no time passes for you, there would be no way for you to decide that you wanted to stop travelling at light speed - you would end your existence by impacting something.
> that certainly makes no sense to me
Humans, as a species, aren't experienced with relavitistic effects; we have no useful inherent intuition as to how things behave at these sorts of speeds. Most of us have to accept the maths and the experiements that demonstrate that our understanding of this behaviour is accurate.
And you having a rest mass cannot ever reach the actual speed of light, only a factor of <1.
If you were to travel with 95% of the speed of light your rest mass would increase by 3.2x ... at 99% by 7.1x at 99.9% by 22.4x and at 99.99999% by 2240x and so on. The Lorentz factor is infinite for the speed of light. Time "stops" and the object contracts to "zero" at c.