Yeah, it's expensive, but the Fuji X100T is the best camera for taking great pictures of your family (buy it from Amazon).
Ask yourself - how much is it worth to have pictures of your kids that just look great, without a lot of fiddling? The Fuji X100T has a huge sensor (even larger than the one on my favorite superzoom), so it takes in a ton of information - that means you can crop a picture if you want and still make a print that's worthy of framing.
The Fuji X100T is popular with photography professionals, because it will do what you ask it to do ... yet if you simply leave it on auto-everything, all you have to do is point and click. You've just become a better photographer, and all it took was a lot of money!
There are a couple of reasons I recommend such an expensive camera for family photos. First of all, you'll never get those moments back ... if you have a new baby, you're simply not going to wish you got a cheaper camera, and you won't regret having pictures that make your friends say "Wow!"
Beyond that, candid family pictures require the utmost in speed and simplicity - if you have to think about futzing with your camera, you tend not to take it out (or you miss the picture). The X100T does all the thinking for you - just point it at your beautiful brood and shoot. Your iPhone already takes great pictures ... to get something better than your iPhone that you will actually use, you need to spend some money.
Note that the X100T has a fixed lens (no zoom) that's pretty wide. That simplicity makes it easier to capture pictures, so it's actually better for most candid family situations (ever try explaining a zoom lens to your parents?) But if you really want a camera with a zoom lens, check out my recommendations for sports, travel, or learning.
The proof is in the pudding - but remember that professionals will make great pictures using a $3 disposable ... so unless you're a professional photographer, you should check out pictures that ordinary people take with a camera. Lucky for us, Flickr has a great feature that lets you do just that. Here are family photos taken with the X100T. Better yet, here are pictures tagged "baby" ... I don't know about you, but I'd be pretty proud of many of those photos.
Here's an example - a random, unedited photo from Flickr taken with the X100T:
This was a split-second moment of adorableness that was undoubtedly soon over. The most important aspect of this picture is getting it quickly. You don't have time to mess with settings or zoom - just shoot! You can always crop out the distracting bits later:
The first X100 is still a very good camera, and you can get one used for about $500 on Amazon ... or you could get a superzoom like the FZ200 - even though it's quite a bit cheaper, it's still takes good pictures and is very versatile ... at the cost of being bulkier and more complicated (again, making it less likely you'll get the shot in time). And of course, you could get my favorite DSLR if you want to get deeper into photography - but let's face it, even if you do get deeper into photography, you're going to want to have this Fuji, too ;-)
Just get it
If you're a new parent wanting to take amazing pictures of your family for the next few years, this is the best camera you can buy. Even if it's not the only camera you ever buy, it will be your favorite. So don't hem and haw ... hop on over to Amazon and buy it.