The Intel Core 2 Quad Q9400 is not the best processor out there, but for your socket (Intel 775), it's one of the best. Don't get me wrong, it's still a strong processor, and performs similar to an Intel Core i3-2100 (the Q9400 is better than my current Sandy Bridge Pentium.)
So for general use and even for gaming, the Q9400 is a great processor and should last you some time. However don't rely solely on your CPU for a performance upgrade. For gaming you'd want a graphics card too, depending on what games your children play. If it's simply 2D web-based/browser/Flash games, then you won't need a graphics card; but if it's something graphically intensive such as 3D games, then you'll need a graphics card. Feel free to list what games your children will be playing, and I'll advise if a GPU is necessary.
Also as mentioned, CPU isn't the only way to improve performance, as you most likely know. You could look at your RAM, and even replace your hard drive with a Solid State Drive (SSD). I recently installed a SSD into my father's laptop, and boot up times have been reduced by over 20 seconds, and software such as Adobe Photoshop and Microsoft Word open in seconds. If you want software to open "fast," don't rely on your CPU alone, you should invest in a SSD. Having a decent processor also will play a part however, and the Q9400 is more than decent.
I hope I have assisted, and if you still have more questions, please let us know.
All the best.