Is it true that Jupiter is so big that it doesn't orbit the sun?

 Yes!

Jupiter, the largest planet in our solar system. It's so massive that 1,300 Earths could fit inside it. Jupiter is 2.5 times more massive than all the other planets in our solar system combined!

This is our little earth compared to Jupiter:

Jupiter is so large that the center of gravity of the sun and Jupiter is outside the sun, although it is extremely close to the sun's surface.

So the Sun and Jupiter both orbit around their center of gravity, also known as the barycenter.

The barycenter is the center of mass of two or more bodies orbiting each other and is the point around which the bodies orbit.

Technically Jupiter orbits the center of the sun and Jupiter is outside the sun.

So yes! Jupiter is so big that it doesn't orbit the sun. Fascinating!

Now, other planets in the solar system are so small that the center of gravity of the sun and a planet like Earth are located inside the sun. (Even very close to the center of the sun.)

So it looks like other planets are orbiting the center of the sun.

Source: Baricentro - Wikipedia

To understand this, you can also imagine turning a child around like this:

(It could be dangerous. Just imagine.)

The center of gravity of you and the baby would be somewhere very close to your feet. So you're spinning almost in the same spot, but the baby is orbiting you. This is exactly what happens with the sun and the other planets.

By the way, this is how two stars with nearly the same mass orbit around their common center of gravity in a binary star system:

Center of gravity - Wikipedia

How does our solar system orbit the Milky Way, and why does it take 250 million years to complete one orbit?

 The solar system orbits the Milky Way galaxy at an average speed of about 828,000 kilometers per hour (515,000 miles per hour), taking approximately 230 million years to complete one full orbit around the galaxy's center. This journey is often referred to as a "galactic year."

If you could teleport instantly thousands of light years outside our galaxy and look down upon it with supertelescoping vision you might see something like this:

The Milky Way galaxy spins like water going down a drain with our local group in one arm halfway between the core and the outside edge at a distance of around 27,000 light years.

Our solar system orbits with the Sun dragging the other planets in tow, making the solar system look like it is flying sideways, always moving up since the solar system is tilted that way.