It will absolutely depend on your personal preference, your budget, what you like to shoot, where you shoot, and how committed you are to improving your skills. And I have recommendations for you!
In general, using a DSLR is the best way to take better pictures: not because the camera itself can magically take better photos, but because a DSLR gives you the ability to control the things that are going to make the biggest difference in your photography. It will be able to freeze motion, which is obviously a big deal for photographing children. Using the right lens on a DSLR allows you to choose what is in focus and what is not, which one of the best ways to create a pleasing portrait. It has a larger sensor, which means your images can be sharper, more detailed, and higher quality than anything a point and shoot can produce.
Have you ever felt like you wanted to take a picture a certain way and your camera wouldn’t let you? It wouldn’t focus where you told it to, or the picture didn’t look like what you imagined? These are signs you’ve outgrown a compact camera and should consider moving up to a DSLR. That doesn’t mean you have buy a camera this instant; but it does mean that you want to do more creatively than your camera is letting you do. Why would you keep using a tool that limits you?
Now there are two things you should know right now:
First, you need to be in at least the $500-700 price range to get a new DSLR camera. You may be able to find used gear for less, but if you’re buying new, that’s the price. Start saving your pennies, because a DSLR is an investment, but it’s an absolutely worthwhile investment.
Second, you need to recognize that a DSLR a creative tool, not a magical fix for your photos. There is a learning curve, and your pictures will not look amazing if you stay on automatic all the time. A DSLR will produce sharper pictures, even on automatic, but at some point you’re going to have to learn how to use that expensive camera if you want to take the pictures of your dreams. You can absolutely capture the wonder and magic of childhood, but you’re going to get out of this what you put into it.