Physics

Yes the concept has been explored but don't let that stop you please! This is called a stellar engine or Shkadov thruster and it is really cool. By the time you're swinging the whole system around like a piñata at the end of a stick you can also move other bodies, saaay suns, and if its millions of years in the future the people in your world probably have the time to spare to go get them, so the sky is the literal limit. Have fun!

Hm, there is the Bowl of Heaven series by Gregory Benford and Larry Niven, which utilises a similar concept. I think you will have to read quite far until the concept is explained, but it's been a while.

There was a PBS Spacetime video that entertained making the solar system into a spaceship - I think it was this one:

https://www.youtube.com/watch?v=kQNqmUZrEOk

Yes, you can do this in theory. The very simplest form of the idea I know of isn't even technically that high tech, just very large in terms of manufacturing scale and also very slow (even by slower than light star travel standards).

You've probably heard of the concept of a solar sail, where a spacecraft with a very big, thin reflective foil sail is pushed by the sun. If you design the size and mass of such a spacecraft right, you can make it's outward acceleration when pointed directly away from the sun equal the inward acceleration due to gravity. Add some means to control the orientation of the sail to keep it pointing directly outwards, and it can be made to stay balanced, staying in one spot, not actually in orbit, just hovering.

Say we make a lot of these. We figure out how close to the sun we can hang them without them overheating, and we cover one side of the sun, almost completely, in these sail craft (the actual amount of material, though large, might not be that insane even, if the sails can be thin enough. Not compared to the mass of a star anyway). They don't have to be attached together into a rigid shell, just a big swarm of them whose net effect is to stop most of the light on one half of the sun and reflect it to the other side. We can leave gaps between along the orbital paths of any planets we want to still have light.

The swarm of sails will stay bound to the sun, and the combined sun-sails system now emits light but biased in one direction on average. This makes it basically a form of rocket. It'll be extremely slow to accelerate on account of the extreme mass of the sun compared to the light and such coming off, but in space even a small acceleration adds up over time.

This is called a shkadov thruster. There are other, more complicated but probably faster to accelerate ways you could hypothetically use the sun to move itself, but I'm less familiar with how they work. The wiki page on stellar engines might give you ideas.

This is probably not a great way to go colonize other planets though. The sheer scale of manufacturing capability needed to do this implies enough resources and energy to launch much faster spaceships, and even a spaceship with spinning habitation rings so big or numerous as to fit an entire self sufficient country inside would be much easier to move around than the sun. You could even use the sun to move the ship instead of moving the sun (for example, take that swarm of solar sails, cover the whole sun instead of just one half, and include solar panels to turn a lot of the sunlight into usable energy. Then run big lasers off the sunlight, and point them at another sail attached to the ship, so that it's like a solar sail but with way more light pushing on it than it would normally have).

You could send out probes to do this to other stars and bring them closer to home though, or even hypothetically send out a very long term mission where a probe that can build copies of itself goes to another galaxy and when it gets there, builds copies and sends them to nearby stars, which copy themselves again, until every star in that galaxy has the probes, and then the probes build the mirror swarms and direct all the stars in the new galaxy to fly towards your original one. It'll take them a very long time, millions of years at the very fastest and closest, to get to you, but "running out of materials to the point that you need to harvest entire other galaxies to satisfy your needs" is such a far off problem that that's probably not an issue.

Yes! There’s actually a fictional universe way ahead of you. See, for example, the Sundrivers, which use their star systems for thrust: https://www.orionsarm.com/eg-article/478adb4aeb392

Or Dyson Beams, used to propel large spacecraft and for communication: https://www.orionsarm.com/eg-article/48fe49fe47202

Planets arranged as phased array communicators/propulsion are quite common.

But there are more exotic solutions here too, like re-arranging stars into more compact configurations, or “metric” engineering involving warping space time in theoretically plausible, non relativity violating ways. This even (theoretically) allows for the creation of wormhole network and highly relativistic spacecraft, though with immense difficulty and complications, and no true FTL.

As an addendum, you actually don’t need to go millions of years into the future! A few millennia is plausibly enough for feats of engineering that are way beyond our comprehension: https://www.orionsarm.com/xcms.php?r=oa-timeline

Orions Arm is basically a wiki and a collection of short stories on this… Not the worst setting to write a novel in.

As an alternative, there's also the idea of collecting a huge amount of solar power to make a tiny black hole using some very very hard to make lasers. A 5×10^9kg black hole is ideal.

Isaac Asimov talks about it a bit in Imperial Earth and from what I remember he uses accretion of matter as a source of fusion energy. Back in my early uni days I did a small research project on this idea, and you can actually get a ton more efficiency by turning the whole thing into something like a NERVA thermal nuclear rocket engine. If you include Hawking radiation in the calculations and are particularly clever with the design, the whole thing becomes suprisingly controllable over a range of thrust, and you get a constant 1g acceleration which resolves a bunch of biological issues around weightlessness.

It does require a little magic. You have to tether the black hole to the ship, and you need to use some pretty intense magnetic fields to contain the exhaust that's being ejected at about 87% the speed of light (this also turns it into a fantastic way to destroy a planet by pointing it at the surface), but the fuel pumps used on the first stage Saturn V F1 engines are enough to feed it. The main issue imo is the absurd amount of radiation that'll sterilise all life on the ship, but you can reach Proxima Centauri in about 4 years and the black hole should last for about 1000 before it's spent (which interestingly is more about it becoming more massive, not less).

What you are describing would probably be best fit by what's called a "Dyson Sphere", which would be a spherical shell constructed around a star in order to capture and make use of all of the radiant energy of said star. Such a construct would have to have the ability to manipulate gravity in order to keep the star in the center of the sphere, and one could extend this concept to include habitable planet(s) in orbit around this star, inside the Dyson Sphere. Now, slap an Alcubierre drive on your Dyson Sphere, and you are cruising yer solar system through the galaxy.