Hi @PhoenixClose1,
A small diameter drill bit such as this Sutton Tools 3mm Viper Jobber Drill Bit would be perfect for a pilot hole for these screws. You may even find that the existing nail holes are enough of a pilot hole for the screws to drive into. A pilot hole doesn't need to be big; you are just clearing a bit of space to allow for the screw.
Having a look at the damage to the fence rail, it has splintered quite a bit. Unfortunately, there is not a lot you can do about this. You could seal the timber with some fence paint such as this Walpamur Charcoal Fence Finish to offer some protection, but once the timber has started to deteriorate, the best option is usually to replace it.
Let me know if you have any further questions.
Jacob
