I love the pictures you posted, no doubt you have a good "eye" and I would say dont limit yourself to just taking photos of your son.
To take good pictures you really need to learn 2 main things.
1.How to use the camera from its mechanical point.
You should learn the basics of ISO, Shutter Speed and Apature.
2.Learn composition like Rule Of Thirds and much more.
Just buying a DSLR and using it on auto is in my eyes a waist of great potential and frankly they are heavy and big and dont forget the lenses you need to buy extra and carry along with you so these are my thoughts..........
If you really want to learn about camera and how to use it then go for a DSLR all current DSLR cameras are good but since you never gave us a price point I will tell you those who I recommend.
For around 1100$ you can buy the Nikon D7000 which is in my eyes the best camera in its price range.
For less then that you can get the Canon T4i or the ceaper Canon T3i
Nikon D5100 or the newer and more expensive Nikon D5200 who is currently available in Europe.
Nikon D3200 is Nikons brand new entry level and is resonably priced and if you really want to go cheap then the Nikon D3100 is a good camera that will be the cheapest from my list.
You should buy these cameras with a zoom lens minimum the 18-55mm or better, you also should buy a prime lens especially if you like taking portrents. 50mm prime lens probably will be perfect with f1.8
If you rather go for something smaller and dont have the time, will or patience to start getting into learning this wonderful world of photography you can get one of the good quality point nd shoot cameras.
Today some of the point and shoot are very close in their performance to the entry level DSLR but without getting into all the extra lenses, size and weight of a DSLR.
As an example let me give you 2 models I would recommend.
Canon G1X has a sensor that is almost the same size of a DSLR and will take wonderful pictures in all condition but the down point of this camera that its not cheap (around 650$)
The second camera I would recommend and thats from my own experience is the Canon G15, I am recomending this camera because I own it and it really is a great little camera which has a very fast lens and is perfect for portraits and excellent in everything else.
It doesnt have a huge sensor like the Canon G1X does but it is still very good in low light and indoor picture and is considerably cheaper then the G1X
Of course their are other good Point and Shoot out there but I did my research and found the Canon G15 to be best value for the money.
So I think you have a lot of thinking and research to do.
Being the father or 2 I know how much fan it is to take pictures of my kids and those pictures of them when they are young will be a memory you will keep forever so choosing the right camera is important so you can take these pictures and they will come out just as you want them to come out.
Good luck
