Is it possible to upgrade the native python version (2.6 on my Snow Leopard) to a more recent one? Or do I have to just install python 2.7 besides the current one?
You can install 2.7.
It is generally considered a bad, bad idea to change the native version, because built-in apps and frameworks are written for specifically the version that is bundled.
The way to go is to install 2.7 along side 2.6. This is generally considered the best practice.
That way, you can use 2.7 when you want, and apps will still use the version they were made for.
Go to this page to get the Python for Mac installer. Follow the directions to install it. Then, to access the new version, type
python2.7 in Terminal. You'll have a Python shell of the version specified.