What you'll be doing as a dad
Holding your baby in a standing position for long periods of time, often while pacing back and forth as he or she screams in your ear; leaning over changing tables for diapering and dressing.
What you need to build
Your lower back. Generally the most neglected body-part in the average dude's workout, those erector spinae muscles will be crucial for all the lifting and bending.
Back extensions. If they're not already part of your fitness routine, start very slowly. Try 3 sets of 8 reps on a back-extension bench, and once you feel comfortable with that, add reps, or try it while hugging a 25-pound plate to your chest.