Which timestep should I use for a $N$-body simulation of the Solar system? I am trying to implement a $N$-body simulation of the Solar system and I am stuck on the issue of the simulation timestep size. At first I would like to use a simple Euler stepping scheme. Knowing that the error is dependent on the size of the timestep, I am not sure what to use for the model.
Would using a timestep size of one day be enough to get valid results? And also I would need to input the timestep in the model using seconds (60*60*24), correct?
Finally, where can I find values for the initial conditions that I should input in the model (initial positions & velocities vectors).
Sorry if these questions are kinda basic but my physics is about 15 years rusty... :)
 A: The standard way to choose a time-step is to run a test simulation with multiple bodies and plot the total energy of the system versus time. The total energy should remain (roughly) constant. If your step size is too large then you will get energy drift. So simply find the largest time-step that does not produce energy drift. In the case of modelling the solar system, my hunch is that a time-step of 1 day would be too large. I'd suggest trialling something closer to 1 hour.
By the way, more advanced codes tend to use variable time-steps, e.g..

And also I would need to input the timestep in the model using seconds (60*60*24), correct?

That entirely depends on the units that the code uses. It's generally regarded as bad practice to use SI units though for modelling astronomical systems due to the huge numbers involved.
You may find the Wiki article on Numerical model of the Solar System to be informative.
A: Direct integration schemes give bad results. You can do much better by using the exact solution in the absence of gravitational interactions between the planets. You can then set up a variation of constants approach where you take the integration constants (which are the orbital parameters) as dynamical variables and write the differential equations (where you now include the mutual gravitational attraction between the planets) in terms of these variables.
Because you now only have slowly varying functions to deal with, you can take much larger time steps. The downside is that the math is much more complicated involving functions that need to be evaluated using numerical integrations. 
