i use a desktop at work, and probably always will because it's big and powerful. At home, my wife and I each have HP "sleekbooks" that are ultra portable, slim, and powerful enough for the work we do at home. We have a big 27" full HD monitor that we plug the laptops into via HDMI, and connect a wireless keyboard and mouse, and essentially have a desktop set up. Then we can always unplug, and go work on the couch. This works very well for us, and gives us lots of flexibility...and I don't think I'll ever go back to a desktop set up at home.
In your case, I think you should get a desktop if you have no plans to be mobile/work anywhere other than your desk, because you can get more computer for cheaper...but if you have any desire to work in different places, or take your work on the road, etc...definitely look at laptops. As others have mentioned, for the type of things you've described, you likely don't need the TOP of the line specs, so you should still be able to find a good deal on a good laptop.
And finally, as with all home computers...my suggestion is to give the kids their own. Whether that is your old one (wipe and reinstall Windows and you may get more life out of it), or their own new system, or even something like an inexpensive Chromebook, having separate computers for parents and kids makes a lot of sense to me.
If you need suggestions on any of the systems/components mentioned so far, just say so and I'm sure plenty of people here would be glad to provide you with links on some good deals. Just mention your budget, wishlist, and maybe what kind of programs/games you use it for.