Humanity builds a spaceship to leave Earth. For eight billion people or so, that vessel has to be gigantic. Using the metal from Earth's core for the construction and the biosphere to create a habitat, building the spaceship more or less uses up all of planet Earth.
But how do you get all the people onto a spaceship that hasn't been completed yet, while their planet is being deconstructed under their feet?
You may use other celestial bodies in the solar system to solve the problem of logistics. You may use other celestial bodies as a resource for building materials, as long as the basic premise of using up (most of) Earth to build a spaceship is kept intact.
The passengers aren't stored but living on the spaceship. They need to take with them everything they need to sustain them indefinitely (as space travel takes a long time). For that, the spaceship needs its own biosphere, and taking the one that has served us well for the past couple of thousand years is probably the easiest strategy.
The whole process of moving humanity onto the spaceship should be completed in a few decades at most. Whatever technology you need is available. Again, this is not a question of technology, but of logistics. "Beaming up" people does not work, though. They have to travel, but you can make this quick and large scale (ocean liner sized rockets or space elevators).
Moving Earth itself doesn't seem an option to me. First, it takes too long to accelerate Earth into a useful direction and a meaningful speed (see How can I move a planet?). Second, the atmosphere would be stripped away once we moved out of the protection of the Sun and into denser parts of space. Third, we would have to move an orbiting source of light along to have a source of sunlight. It seems easier to have a light source on the inside of a vessel.
I'm not saying that Earth's biomes have to remain unchanged. I want to move humanity, and to do so we need food, air, etc., all of which are easiest to produce on something like Earth's biome. I mean, you could probably synthesize proteins etc., but why not simply grow wheat etc., which would also solve the oxygen problem? Also, some people need "nature" to remain psychologically sane. Not everyone is made to live inside a room and never be out in the open. So recreating something like the surface of Earth seems necessary to me. But we may not need to have the same kinds of plants and animals, not all of them, not the same kind of landscape, etc. The goal, again, is not to move Earth, but to move humanity. If you can do that by moving Earth, fine, but it is not a requirement of my question.