I've always carried a small backpack with me pretty much everywhere I go. This is perhaps a hold-over from school and uni, continued because I always want to have a couple of books on me. These days, however, there's a lot of other stuff in there as well.
First of all, cycling around Oxford in winter means I'm carrying:
* a scarf
* a beanie
* gloves
* a bright yellow cycling shell - windproof and sort of waterproof
* nylon overpants (when rain seems likely)
* a bicycle seat cover (when rain seems likely)
* front and back bike lights
Then there's assorted electronics
* my iPod touch
* my little camera
* my Kindle
* my bank security device (aka a mobile phone)
That still leaves room for a couple of books, pens, boxes of tissues, and so forth, along with some spare bags for shopping. I usually do larger grocery runs on foot, so I can carry a couple of separate shopping bags, but I should perhaps get a set of pannier bags for that.
Soon, you might need to fit a rattan seat on the back or handlebar of the bike for the baby. You'll have to find room in your pack for nappies, bottles, and thermos of hot water. That's just the beginning.
Anna seconds that on FB with "You'll soon be adding wipes, nappies, a change of clothes and even a squeaky toy or two". But I hope that will all fit in the pram!
It's not advised to take a baby on a bike till they're 8 or 9 months old and can hold their head up http://www.guardian.co.uk/environment/2009/aug/12/baby-seat-bike-blog but once that time comes I think I'll definitely need pannier bags, or maybe a rear basket to go with the front-mounted child seat.
Or there's the bakfiets option: http://babyccinokids.com/amsterdam/2010/02/02/do-as-the-dutch-do-bike-on-a-bakfiets/
Funny thing, I thought I was the only one carrying all sorts of things on a daily basis.
In a koala-like bag, I carry a small first aids kit, a water bottle a multitool and a small torch (plus other junk, as required).