Forget the bulb thing that they gave you in the hospital. It doesn't work that well. Those things always made my baby gag and throw up.
Use a q-tip. Not the safety q-tip that Johnson and Johnson makes because it doesn't go deep enough
Take a regular q-tip and place it between your finger and thumb and swirl it, so the cotton is tight on the stick.
gently put it in the baby's nose to the side
swirl around and pull it out
This has always worked for me. When he lets me clean out his nose. Babies definitely hate having their noses cleaned, but this is the easiest way.
Use the q-tip when he is taking his bottle or playing with something so that he is distracted. Be sure to focus and be careful, because if he jerks it could be painful for him.
Good luck!