Racedude,
I'll try to give you the best answer I can.
First, I think that there is a mix of VFR (your aircraft initially) and IFR (the other aircraft on the ILS and your following the SID).
Scenario 1: You are VFR and the approaching aircraft is IFR
If you are flying VFR, in the real world, before taxiing, you would give to the tower / ground your direction of flight to exit the airport airspace (most likely a class D). In this case, it would be a south west departure. Again, assuming that the airport is in a class D airspace, the controller will take care of you within a 4nm radius (typically): there is "no separation service to VFR aircraft" within this radius, so, in VFR, YOU (the pilot) are ultimately responsible for traffic separation.
As far as I know, the controller will try to avoid you crossing the path of the ILS unless you fly jet and can be way higher by the time you cross it.
Then, once ready for departure, they would clear you as follow: "[calsign] clear for take off runway 05, RIGHT turn on course". You then take off, climb to 500ft AGL and start your turn to the southwest, remain south of the runway and maintain heading until you leave the airspace. When you are out of the airspace, the controller "doesn't care" anymore about you: you can "resume your own navigation" and change your heading to reach the starting point of the SID.
Scenario 2: both aircraft are IFR.
Then, your SID specifies which direction you should turn. You would also have a clearance and the controller would be aware of the path you are following. Then, he would either clear you to follow the SID as published or would vector you if he thinks that it might conflict with other traffic.
Finally, I do not fly P3D but if it is like FS, don't rely to much on the ATC service: it is extremely basic and definitely doesn't cater for such cases. You may try the on-line service Pilotedge for more realistic procedures (full disclosure: I do not work for Pilotedge neither receive money from them).
I suggest that you:
looked for your airport on Skyvector: you would be able to find the complete path (from the runway) of the SID at the airport you use. It is very unlikely that the SID would have you cutting the ILS path.
watched the video from Pilotedge: there is a complete workshop about IFR departures (and generally other videos about IFR procedures).
read the FAA "Instrument Flying Handbook" (available in pdf format for free) for a very detailed answer.