It’s not impossible for things to move at the speed of light. They can and do – light, for instance. And in such cases, no time passes for them on their journey. But only things that have zero rest mass can do this. Everything else is confined to move at speeds less than light.
If we wanted to accelerate an object with non-zero rest mass up to light speed we’d find that, no matter how much energy we used, we’d just get ever closer without ever reaching the speed of light. Even trying this with two objects and sending them off in opposite directions, and hoping their combined relative velocities would do the trick, wouldn’t work. This is usually offered as the reason why ordinary objects cannot be accelerated to the speed of light - that an infinite amount of energy would be required.
As you say, as things move relative to us, the rate at which time passes for them, as measured by us, slows down. But that’s only as measured by us. To observers traveling with the ‘moving’ object, their time is running at the normal rate, and indeed, if they measured the rate at which time was passing for us, they’d conclude that our time was running slowly.
If we allow ourselves a little imaginative license (well, OK, a lot), suppose a photon of light could carry a wristwatch. The hands on its wristwatch wouldn’t move at all between the start and the end of the photon’s journey. While we would measure the journey to take a finite amount of time, the photon would experience no time at all passing. So photons arriving now on Earth from the Sun have been traveling for over 8 minutes, as measured by us, but to them the journey has been instantaneous. Persisting a little longer with the ridiculous notion that photons could experience anything at all, they’d also notice that the distance between the Sun and Earth had contracted right down to zero. So no wonder it seemed to them to take no time at all to make the crossing. And this, of course, applies to any photon journey.
