How could a Kardeshev II level society move an earth-sized planet at high speed, without killing everything on the planet and access to all of the resources in a solar system similar to our own? what is the shortest timescale this could be accomplished on?
This is covered pretty well by How to move the Earth blog.
The only extremely remotely feasible way is a repeated gravity assist. Essentially, you take a big asteroid and have it make a flyby of Earth, transferring some angular momentum from itself to the Earth, then make a flyby of Jupiter, transferring angular momentum from Jupiter to the asteroid, and repeat this billions of times. The Earth orbit will get ever so slightly higher, however it certainly doesn't move fast. As described by the blog:
This is a method originally proposed as a means of moving Earth to a higher orbit around the Sun in order to save it from the Sun's inevitable Red Giant expansion. It involves asteroids, like the above method, only instead of direct impacts, this time we just steer them past the Earth, allowing rock and planet to exchange a little momentum, with the result of an Earth moving on a slightly different track and an asteroid moving on a significantly different one. You could reuse the same asteroid again and again, looping it around a few gas giants and back to gain lots more kinetic energy from those gas giants in the same way that Earth just gained velocity from the rock. You could repeat this thousands of times over the course of millions of years. Better, you could use many, many asteroids one after the other in a steady stream, and cut down the total time significantly. You could of course use this method to steer the Earth in any direction you wanted, not just away from the Sun... heh heh heh...
Divide and conquer.
Build a fleet of enormous space ships and fill them with all of the fragile parts (humans, animals, plants, cities, etc.).
Next use your K2 technology to condense the atmosphere into an liquid form and tank it up, along with the oceans, in secure storage tanks on the planet's surface. Then solidify the core and unify the continental plates, reducing the planet into a single, high density object capable of surviving high speed acceleration.
Now the barren solid planet can be pushed and/or dragged through space using K2 level gravity manipulation. By constantly creating a intense gravity well just ahead of the moving planet, the planet can be kept in a constant state of descent, increasing in velocity until it reached some physical maximum, limited only by its mass.
Meanwhile, the fleet of spaceships can escort the planet or cruise along at a more life-friendly velocity. After all, it is not a problem if the planet gets to its destination ahead of its inhabitants.
The goal was to move the planet.
That depends on what constitutes "high" speed, whether you need high acceleration or just high final velocity, and where you intend to move it.
Most viable options are going to involve some sort of gravity tractor. Maximum acceleration will thus be limited by the gravitational force between the planet and the tractor, and thus by the mass of the tractor. Moving faster will thus require a heavier tractor, which will require more power to move the tractor in addition to the planet it is towing.
Your best bet is probably to build a high-mass annular tractor in orbit around the planet. That'll let you make the tractor comparable in mass to the planet itself, or even more massive, while minimizing the impact of the tractor's tides on the planet's surface. It's not as ideal for that purpose as a full shell... but a full shell wouldn't be able to tractor the planet at all, so an annulus is probably the best compromise. Then, just blast the annulus with laser light from your Nicoll-Dyson sphere, or your stellar coronal gas lasers, or whatever megamachinery you are using to channel the full power output of a star, and make sure it's angled to reflect said light in the correct direction for the thrust vector you want, while being careful not to let any of the laser light shine on the planet itself.
A safer option with a lower mass requirement, but also a lower maximum acceleration, is to use a mass stream. You build a few trillion asteroid-or-smaller-sized drones, equip them with light sails, and set them on slingshot trajectories around the planet in a continuous stream, relying either on regular solar sailing or active laser guidance to reset each drone for the next pass after it has passed the planet and transferred its share of momentum.