You can't go through the centre of the sun with anything like present day technology. Denser than platinum, a billion times the pressure in the ocean depths, high gravity - only way to avoid the gravity is to go through it at great speed so you are in close to a free-fall path. But - h0w can you travel at great speed, many kilometers per second, through material that is as dense as platinum? It's also so hot, that even a spaceship far more massive than the sun would melt through before you got through.
The outer layers of the sun however are very thin, close to a vacuum, thinner than Earth's atmosphere. So you could travel through those outer layers if you had a very efficient way of reflecting the heat away or great refrigeration. So your spaceship could seem to "dip into the sun" - but actually it's only dipping into the outer layers of it.
It's not in the way anyway, for travel from Earth to anywhere else - it's huge compared to Earth but tiny compared to Earth's orbit, and it takes far more change in your speed to go close to the sun than it does to travel around it at a distance - it is as hard to go deep into a gravity well as it is to escape from it. You can't just "fall into the sun" because you start off in orbit around it at many kilometers per second velocity.