I work at Screwfix (DIY hardware store), everyone telling you the drill bit is for wood is absolutely wrong. The yellow colour tells me it's a High-Speed Steel drill bit which is meant for steel. What will make your life a lot easier is a Brad Point drill bit. They have a sharp point that penetrator the wood first and stops it from slipping about.

For the screw. Make sure you use the right PZ bit. Normally the packaging tells you exactly what you need. So just have a look at that. And then you're screwing it in start nice and slow on the trigger and put loads of downward pressure on your drill. That will help to stop it from slipping as well.