Hi, Lisa. John 2000 is right that you can use a screwdriver. Don't even think of using my chisels for this. But there is an easier and simpler solution. At your home center you can a flat pry bar. Mine is a Stanley "Wonder Bar" and it works great for this and a lot of other demolition projects. They are not very expensive.
The best wood filler that I have found is from Minwax. I forget the exact name but it is something like wood filler. This is a 2 part product with resin and hardner. Since you have small holes to fill, only mix about half the suggested quantity at one time since it sets up fast. This would be half of a golf ball size glob of resin to 3/8" of hardner. No wood filler that I have found takes stain like wood so you might want to try daubing a darker version of your stain or wood color on the wood filler before refinishing.
Hope this helps. Let us know if any more questions. Henry in MI