Driving with an auto in manual mode is no different than driving with a stick other than a couple of things.
The auto will upshift when ever you tell it too but will only down shift when it is safe, it will not let you over rev the engine. Also it will hold the gear your in and will not upshift, it will bounce off the rev limiter or fuel cut off and not upshift so you can't hurt the engine. It will not let you down shift if it will over rev the engine and also has rev match built in and will blip the engine when you down shift.
As far as making the car accelerate faster it's because it doesn't upshift to soon and you control the shift timing. There is a delay when the shift happens after you shift and this can be reduced with a throttle body upgrade. When you control the shift time you can control the RPM your engine is running at and keep it in it's power band for better acceleration.
Just practice a little with it and you will be fine and don't worry about hurting it as there are plenty of safety controls built in.
I have a 2006 G35 that I take to the track and always you my paddles to shift. Here is a link to a video and you can listen to my shifting.
https://www.youtube.com/watch?v=XedOmuTbTck
Hope this helps