Always ensure that your baby had enough sleep the night before going out and feeding him/her just before going out will cut down on the extra weight of water and milk bottle that you need to bring along.
-An extra set of clothes for your baby in case he/she stained their clothes
I bought this "pig" hand-puppet from DAISO for only $2 and it's really helps when it comes to providing entertainment for my baby, especially on boring train rides or waiting for bus.
I don't have a REAL diaper bag, I would just use my normal bag and because I don't bring a lot of unnecessary stuff, they all fit into the bag quite nicely and I even have space for my own wallet, phone and keys.
Instead of pushing a pram and having all the trouble to open and close the pram, let alone carrying it up and down the stairs, I opt for a baby-sling. I am going to be honest, it bloody hard to use initially but after many practices, you will get quite used to it. And I appreciate the fact that my baby can drop off to sleep quite soundly in it. But still I wouldn't give it 100 points because the sling tends to slide off and needs constant adjustments.
1) If your baby had finish all the food/milk and is still crying loudly because he/she is hungry. Buy a piece of fruit or bread and break it into small piece and feed he/her with it. That usually shut them up.
2) If you run out of hot water to make milk, go to McDonald's, order a drink for yourself and ask them for a cup of hot water.
3) If your baby pour water on himself/herself accidentally and you didn't bring extra set of clothes, take off the wet clothes immediately, squeeze out as much water as possible. Head to a toilet with hand-dryer, blow dry it there.
4) If your baby screams non-stop in the public, put a piece of snack into his/her mouth. And if that didn't stop him/her, just smile at the annoyed faces that stares back at the both of you.
When all else failed, speed-dial the number to call for a taxi and head home immediately! LOL!
For me, I guess the key thing to deal with the awkwardness that your baby can bring you, is to tell yourself that you are blessed to have a baby who is healthy enough to create troubles for you, so just sit back, relax and enjoy the torture your baby brings!