Here's my trick to starting a corkscrew so it goes straight down the middle of the cork. Point the tip straight down, right in the center of the cork and push it in slightly. Straight down is critically important. Even though the rest of the corkscrew will be at a steep angle at first, as you turn the corkscrew clockwise it will turn itself straight up.
Keep turning until the twisty part is about one turn from being fully inside the cork. That should normally prevent poking a hole in the bottom of the cork (and getting little cork pieces in the wine) and will position the corkscrew with maximum leverage to pull the cork out.