I think the diaper bag thing really does vary a lot depending on how you get around with the baby. I'll share my experience, since it was almost completely NON car-centric (and I see that you're looking for ideas that will work for public transit use, etc).
For my baby's first 3 or 4 months, I used a wrap exclusively to get around with her. I carried her on my front, and I used a backpack for our stuff. I didn't have a diaper backpack, but instead just kept my contents bare-bones, and I eventually figured out little tricks to make it easier. Honestly, the main reason I never invested in a diaper backpack is that I've never found one that I liked the looks of... If I'm going to spend money on a new bag (to replace a perfectly functional backpack I already have), then I want to enjoy using it and looks are a big factor there.
Anyway, I ended up buying a stroller when she was 3 or 4 months old and it was a decision I never regretted considering our car-free lifestyle. We continued to use the wrap (and later a Boba) till she was 2, but the stroller was a game changer for me in terms of how mobile I was able to be when alone with the baby in the city, and how useful I could be (shopping for groceries especially). I had a Skip Hop brand Via Messenger diaper bag to use with the stroller and I LOVE that thing. The many pockets are nice, but honestly they aren't a huge deal because I was never carrying THAT much in the bag. I mainly just loved the great clips that let me attach the diaper bag onto the stroller. This left the stroller's storage basket free for other stuff (groceries, library books, etc), and it made the bag really accessible.
Without further ado, here's what I remember carrying in my diaper bag (backpack or messenger bag) with DD:
Diaper changing - diapers and wipes get placed together in a gallon size ziplock bag or small washable wetbag.
2 or 3 disposable diapers - I used disposables when out for a longer period of time, cloth for short trips. But always disposables in the diaper bag because they take up so much less space and they weigh so much less.
ziplock bag of disposable wipes - I need the wipes to be wet for diaper changes or cleaning up messy hands/faces. If I put pre-moistened cloth wipes in the diaper bag, then I have to remember to remove them after each outing. I'm too lazy for that. You could carry a spritz bottle of water with you to wet the cloth wipes on the go, but for me that's just another thing to fiddle with, and another thing to potentially leak in the bag. So disposables it was.
Changing pad - to be honest I often forgot to put a changing pad in the backpack, and I'd end up using a flat diaper that I kept in the bag (see below).
Change of clothes - This goes in another gallon size ziplock bag or small washable wetbag
The change of clothes consists of a complete seasonally appropriate outfit in a size that fits. Socks, pants, and shirt.
Burpcloth
This was always a prefold or flat cloth diaper for me. Early on this was to clean up if she spit up. Later I still found it useful in the diaper bag for cleaning up big messes or wiping rain/dew off playground equipment.
My stuff
Wallet. Phone. Keys.
Drink
A bottle of water for myself.
When DD was about 10 months I started carrying a leakproof straw cup of water for her. (I had good luck with the Nuby brand of straw cup.)
Food
DD was breastfed and I didn't need to carry any special food for her till she was old enough to consider it an interesting diversion... Maybe 10 months? I like the baby food pouches because they can stay (unopened) in the diaper bag forever without going bad. We also do a lot of dried fruit, pretzels, and crackers. Non-messy things.
I like to have some fresh fruit in the bag, but you have to be careful not to forget it in there for too long! Ew!
Extra clothes for a change in weather (not always in the bag - totally weather dependent)
A light blanket, hat, mittens, leg warmers, sweater, umbrella - anything like that that you will use for yourself or the baby if the weather turns crummy.
My DD always loved sunhats and sunglasses, but never actually wore them for long, so I usually have these things floating in the bag during the summer.
In summer, a swimsuit, sunscreen, and travel towel. And once she was walking well, a pair of water friendly shoes. Our neighborhood parks all had sprinklers and kiddie pools.
Toys
I'd use a pacifier clip to attach a little doll or other toy to the stroller strap or the baby carrier when she was smaller. Once she was around a year old I'd maybe throw a board book in the bag, at most.
Oh yeah, she did use a pacifier for the first 6 or 8 months. So I'd keep a spare in the diaper bag, and I kept one pinned to the carrier or to her shirt (attached with a short ribbon) when we were out. This was mainly relevant for her first couple of months.
Sand/water toys - when going to the playground, I have a separate bag of toys that fit in the diaper bag. Shovel, bucket, funnel, basic stuff like that.
Extra room for the things you accumulate while you're out
It's a good idea to leave room in the bag to hold whatever you get on your outing. For me this was usually library books or farmers market stuff.
Things I tried carrying in the diaper bag and found I never, ever used:
Hand sanitizer
Diaper cream
Lotion
Basic first aid kit
Things I think would have been useful, in retrospect:
Sunscreen wipes (instead of carrying around a tube of cream)
Disinfecting wipes for cleaning scraped knees
, DP's only request is the bag not be pink 
... Okay, I can do that. Although now that I see those bags Bena linked to, I may be changing my mind!
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