When I leave the gate and taxi to the runway, my aircraft comes to a stop in a straight position, but when I engage EPR, the plane begins skidding all over the runway and sometimes doesn't even take off.
Are you setting up the CDU/FMC before takeoff? I always enter the V2 speed (from the Takeoff Page in the CDU) in the MCP and I've never had a problem with the 767 not having enough power to take off.

Do you use a joystick/yoke to increase the power before pressing the EPR button and then pull the throttle lever(s) on the joystick/yoke back to idle? If you are pulling the throttle lever(s) back to idle after pressing the EPR button, this will initiate an aborted takeoff - the autobrakes are applied and the spoilers deploy (if you are arming the spoilers before takeoff), bringing you to a stop. When I'm taking off, I line up on the runway, do the before takeoff checklist, the press EPR. I do not use my joystick throttle to start the takeoff run, because at some point I will puul it back to idle and the 767throttles will follow, unless I disable the joystick using the default keyboard command - CTRL+K. A note on arming the spoilers before takeoff - once the throttle levers are more than 20% you can arm them then. Arming them while stopped will most likely cause them to deploy immediately, which may be prevent you from taking off (the 767 might think it's an aborted takeoff).