Global 6000 Pilot reported on departure not having the automation configured correctly resulting in temporary loss of aircraft control.

Narrative

I was the pilot flying. Before take off I selected heading mode before runway alignment. (Departed Runway XX; ZZZZZ departure). After take off I and beginning the initial climb to 3000 I failed to rotate high enough and bring the line" to the box indicated on the PFD (Primary Flight Display). This caused the airplane not to engage the auto throttles as they were programmed. Auto pilot was engaged appropriately 2500 feet. Departure instructed us to turn right 360. Due to engaging heading prior to taking the runway This caused the aircraft not to turn on the assigned heading in the correct direction after ATC assigned us 360 on the heading. Noticing this I disengaged the auto pilot to turn right. The auto throttles were engaged and at max power due to not being engaged properly after take off (failure to rotate the line to the box). Auto pilot was off and a climb was inadvertently happening. We deviated latitude by at least 400 feet. Also after disconnecting the auto pilot to make the turn due to what perceived to be urgency from ATC there was a bank angle of nearly 45 degrees. The aircraft did not say "bank angle" but the turn was aggressive. Realizing the altitude was high; I began correcting putting the airplane nose down and the max power increase where the auto throttles were set at caused the airspeed to be nearly 300 indicated. I disengaged auto throttles and deployed speed brakes to slow down. Eventually the we were able to bring the aircraft under control and resume a normal flight. The pilot monitoring did a great job in assisting. Realizing something was not correct and I was definitely task saturated by the situation; he coached and announced suggestions to aid in safe maneuvering when issues were noticed such as bank angles and altitude. During all of this ATC did not alert us of excessive airspeed but only altitude and heading.This was combination of lack of proper understanding of the avionics system; lack of awareness during flight; low time in airplane and pilot error. What could have prevented this was understanding what failure to accomplish or what did get accomplished at certain segments of the flight can impact the system and the plane not do what you are expecting it to do. I am not sure why I did not rotate to the box. I do not believe I have ever done that. The heading selection before takeoff was a habit from previous airframes and also flying different aircraft currently. I thought I was staying ahead of the game by doing so. I'll be more critical of what is going on with the inputs and corrective actions that need to be taken with automation off. Also selecting NAV Mode may have been a better option as opposed to flying the runway heading in HDG (Heading) mode would have greatly relived the turning issues we were facing when doing the ZZZZZ departure. During the situation I realized the airplanes performance ran away from me and I was not in control as I should have been. I would describe it is as frightening and a big reminder as to what to look for so that this is not duplicated again."