I'd go for for something like a decent, used D3xxx something again (or Canon/Sony/Pentax equiv - whatever feels right for YOU to hold). It's what I started out on (well first dSLR anyway). Simple but great cameras.
Also, just be aware that it is very addictive, and YOU WILL want to buy more glass (probably a 35mm 1.8 within a year), flash, a bag to carry it all in, a little tripod lol
So try some and think about what feels good in your hands whether it is Canon/Nikon etc, they are all much of a muchness. But also try and think about where you are going.
Babies dont need fast focus (not really), kids at park, you want quick response from shutter, requirements can change with age of the kids.
People go on about buying into a system, but i've never seen the reason why you cant change easily enough in a year or 2 as things can be sold on anyway, yeah, you might lose a bit, but thats called depreciation with any consumer goods (not the glass so much).
I'm talking Nikon here, as i've only had Canon P&S, but the 18-55 kit lens is actually pretty decent. Play around with this, get an idea of what you like and the conditions you shoot in often and by that time you will have a list of things that are frustrating you/limiting you in with your current kit, then you look to upgrade (you lack focal length?, you doing more indoors? etc). Then you remove cash from your wallet, get frustrated with your kit and buy again
I've just bought my first compact for years, a Fuji X10. Startup times and low light have came on a bit, but for kids, I am already very frustrated with shutter lag. ie. obtain focus, all good, snap...what you get is a picture of your kids a second or 2 after your snap. May not sound like a big deal, you wait till they are moving fast or even on a swing at the park and your head will be done in with it. dSLR still leads the way for this stuff IMO - although I hear Nikon 1 are good for this (they have other things I dont like so never bought one). X10 good for a lot of things though.
Some may disagree with the above, especially the shutter lag stuff, but its what works (or doesnt with ME), that's the point I'm trying to make for newbie. Also if you go for a D3xxx something or other, you will probably find these days that it isnt the camera that will limit you first but lens specs. ISO, fps, MP, DR are all great these days and we take them for granted at times on any modern dSLR IMO, a lot dont even get 50% features out of their camera body before moving on.
Any camera will get you OK snaps these days, just if you are happy with that and depends where you see yourself going in the future, but that's pretty hard to figure out until you start ($ lol). Buy used. You will want to change stuff around. If its just decent snaps half the time, then just pick up any compact. Yeah, you'll miss some shots at times, but not all...
Knowing what I know now, I would get:
1 flash (this will change your photogreaphy the most IMO if you want to that is) - Nissin £60
1 35mm 1.8 - £100 (cant beat it for newborns around the house, some on here have studios all well and good, but being practical here....)
1 normal zoom (comes with your camera and will be fine)
1 D3xxx - I wouldnt even be scared to look at the D3000 (ok you mention video so maybe not)
35 1.8 would be my first move for your newborns btw.
...and buy it all used. That's what I would do anyway. Lot to take in though and lets face all takes time to learn, but all good fun.
