Getting to an asteroid has many steps, but broadly speaking we can break it into two steps:
The first step is well understood and is now done by a number of commercial companies. There is even a website called Space Flight where you can book your trip to LEO!
The second steps depends on which technology you are using to get to the asteroid. There are two choices:
Traditional chemical propellant method
This method is usually faster, but it uses a lot of propellant. It allows us to use the Hohmann transfer maneuver, which is the fastest way to shift between two orbits. The trajectory for this type of trip can be estimated using the Trajectory Browser from NASA.
Ion Thrust Engine
This is the most propellant efficient way to get to an Asteroid, but it takes longer than the chemical propellant based method. Recent trips by Japanese missions, Hayabusa and Hayabusa2 have used the Ion Thrust engines. It took 3-4 years for Hayabusa2 to get to the Asteroid. Here is the trajectory taken by Hayabusa2.