I know the long term outlook is quite difficult to say, but how close will the Tesla Roadster actually get to Mars on it's first orbit around the Sun?
The closest approach will be on June 10, 2018, and will be approximately 0.74 AU away. So not very close. Per http://www.whereisroadster.com/close_approach.html
But it is of some note that on October 10, 2020, Starman will only be 0.05 AU away, close enough to be affected by the gravity of Mars.
Here is a simulation generated with JPL Horizons data from today (Feb 13). It shows you distance and speed relative to Sun, Earth, and Mars.
By pressing T on your keyboard you can speed up or slow down the simulation, but don't exceed 8192 or the integration will become unreliable.
The trajectory is based on astrometric measurements as recently as yesterday. Horizons warns "Over time, trajectory prediction errors could increase more rapidly than the formal statistics indicate due to unmodeled solar pressure, thermal re-radiation, or outgassing accelerations that are not currently characterized but may exist."