Aerodynamics are less important on a hill, which means riding in the drops or on the hoods with your head low is not as advantageous. When climbing, it's common to sit in a more upright position with your hands on the handlebar tops. Scooting to the rear of the saddle will also help you recruit more of your glutes and hamstrings--the opposite of the forward, aggressive position on the nose of the saddle that may be your go-to when speeding along on the flats.
To get used to this position and use the muscles you'll need for climbing, it's a good idea to dedicate a portion of your training time riding in a more upright position. While you can definitely do this on the road, riding indoors on a trainer is a good idea too.
Use a riser block or a stack of telephone books to elevate your front wheel as much as possible. Spin in a lower gear, around 75 rpms, and practice riding with your hands on the bar tops, sliding back on your saddle and concentrating on lowering your heel into the bottom of the pedal stroke and pulling up on the upstroke to work more of your hamstrings. This will help you build strength and get used to the climbing position.