If you apply a constant force to an object, you will get constant acceleration given by Newton's 2nd law. F = ma, where m is the mass of the object. In principle, any amount of force will get an object moving as fast as you need if given enough time. In practice, you want it done in some finite amount of time. Under constant acceleration, Δv = at. So if you have change of velocity Δv, in mind that you want to achieve in some time t, the force required is given by F = mΔv/t. In the process, you will expend at least E = (1/2)mv² of energy, which will be equal to the kinetic energy of the body. If you plan to use rockets, or something like that, you will use up a lot more energy, but in space, you rarely have a way to get around that.
For orbital motion, there are many nifty shortcuts. If you outline what you want the final problem to look like, I might have better suggestions.