Similar to the Death Star(s) depicted in Star Wars universe, my question is how do such a massive body move from its orbit around its parent planet across the stellar system or galaxy into the enemy's territory? Or perhaps its main weapon is for extreme long range target? How does it defense itself against threat en-route to its destination? I hope it doesn't turn out tragically like in the film! Bonus Question: What benefit does it gain from rotating on its axis artificially?
The Death Star generates artificial gravity by spinning, making it a bit more comfortable to live aboard it. The center and polar regions would still be zero-g or very close to it, allowing for recreational facilities.
A spinning facility like this offers different strengths of gravity at different latitudes, which is a major advantage when it comes to industrial activities. Very large components would be assembled in the polar regions, where the weight is a minor issue than it would be on a planetary body and on the equator.
Variable gravity would also mean that we could vary the degree of convection currents in molten material, which in turn would allow for a much purer material separation processes. Perfect crystallization processes would be possible, which would mean a lot for microelectronics and solar cell development (although I suspect that other applications would be more relevant at this stage of technological development).
As for propulsion, you will - as always - be looking for the most bangs for your bucks in space, since bringing fuel is costly. But for an object this size you would want to focus on the highest possible thrust generating propulsion methods as well, and they should also be able to fire for as long as possible, which narrows down the alternatives. It was probably not a coincidence that Frostfyre mentioned just ion thrusters, since they are among the most efficient propulsion methods we know about and are able to thrust for months or years, with current technology. Generating enough thrust with only ion thrusters is definitely not possible when it comes to moving an object of this size given the tech we have today, but some ion thrusters generate thrust depending on the energy input, meaning that a high producing energy source would allow for high efficiency and higher thrust (I have no numbers though).
It seems very energy inefficient to move an object this size out of a planetary orbit in first place. If you need it in a stable position close to a planet, maybe L5 libation point would make a better alternative since it is a much more shallow gravity well (not sure how stable such a large object would be at that point though).
One of the highest thrust generating propulsion methods we know of is nuclear fission propulsion. The Daedalus project is calculated to be able to generate around 10 to the power of 12 N, as compared to 10 to the power of 7 N for liquid and solid fuel rockets.And nuclear propulsion would allow for years of accelerating, compared to minutes for chemical rockets. But you would still need massive amounts of fuel, and the math is not encouraging for you when it comes to nuclear propulsion. The Daedalus project, for example, was a theorized interstellar ship weighing in at 54 000 tonnes. 50 000 of these were fuel. Yet still it was "only" capable to move at 0.12 c after 3.8 years of acceleration (starting out in Earth orbit). The same goes for a more advanced method such as nuclear fusion. Nuclear reactions release only a fraction of a percent of the rest masses of the nucleii, meaning that any object propelled by fission or fusion would have to carry many thousands of times the mass of its payload in fuel.
Antimatter-matter annihilation is much more energy efficient, but there are some major obstacles when it comes to harnessing all this energy and using it to generate impulse. See the article about Black Hole propulsion below for an overview of different propulsion methods.
The most exotic of all non-FTL methods seems to be the BH propulsion. But again, the enormous mass of a Death Star is not in your favor. A BH with a lifelength of 100 years would - at an efficiency of 10 % - take 15 years only to accelerate its own weight up to 0.1 c. Imagine the energy (and time) needed to accelerate your entire structure to any reasonable fraction of c during the interstellar voyages.
I am of the opinion that it would require some "magic handwaving" to get the equation of a planet orbiting and stellar travelling Death Star together, given the amounts of fuel necessary to get the delta-v's needed, and the energy requirements to produce all this fuel, not to mention the thrust needed to put all that mass of fuel in motion together with the actual payload.
But don't take my word for it, calculate the mass of the structure, and calculate the delta-v needed to leave orbit around a planet of your choice. After that, you calculate the mass of the fuel needed to create this delta-v for an object with your mass, and you add that to your numbers. It would mean that you need even more energy, since you now have more mass, and therefore you would need more fuel (even more mass) to generate enough momentum. I suspect you would end up with a ridiculous ratio of fuel-to-payload for your Death Star - so magic handwaving is probably your best option to keep this idea alive.
Given the massive distances involved, the only realistic method of moving something that large is via FTL which will involve some elements of Handwavium. Even moving from one planet to another would take months or even years!
With regards to rotation, rotation on an axis is often used in harder science fiction as a means of providing gravity. Rotation would also allow for things like solar panels and farms to catch the sun, and for habitat areas to have a "Night" and "day" cycle without too much lighting interference.
First of all, I'd construct the thing in a highly elliptic orbit, and utilize the gravitation of larger bodies to sling-shot yourself. That will actually use your station's own mass to assist in the maneuver. As far as generating enough momentum, I'd shy away from conventional thrusters altogether. For such a big, ambitious space station, you're going to need an equally big and ambitious source of power. Something like a small singularity, or even build your station around a neutron star or somesuch mass. You'll need, perhaps, magnetic clamps (which will probably require more energy than you're likely to get from the results anyway) You'd want this mass to be spinning internally, and maybe even off-center, converting the mass into 'flywheel'. Momentum could then be generated by transferring angular momentum into directional momentum. Assuming you don't rend your own station asunder, you probably won't be very moving very fast at all, so you're going to have to settle for arriving at your enemy's solarsystem about a million years later :/ Uhg, real nerds hate starwars.
The energy needed to debond an earth-like planet is discussed here on this SE, and more exacgly here, and works out to the entire output of the sun for a week just to minimally break it up but not disperse it.
The recoil would accelerate the Luna-sized Death Star to 78 km/s.
So, given the energy requirements of the primary feature, a "photon rocket" can easily be used for thrust, just as a fire-fighting pump barge is a jet boat.