ex matelot,
I figure the python option will at least give me a grounding in the "grammar" of programming.
Up to you really as to how much time you want to put into it.
Being a high-level language, and therefore providing a great deal of abstraction over what's actually going on behind the scenes, a scripting language such as Python will certainly enable you to hit the ground running and make swift progress in your learning.
C/C++ are lower level. They abstract the hardware level somewhat, but leave a lot of the rest of the power in developers hands. As others have pointed out, high-level languages such as Python, PHP and Perl are all written in C.
The other important thing to bear in mind is that if you start off in a high-level language, you're going to find it tough going to wrap your head around lower level concepts at a later date, because you would have got used to the way high-level languages babysit you and hold your hands.
Then again, learning C/C++ is tough going at first. So you may run the risk of being alienated and running away from programming altogether if you're not prepared to persevere. So it may be potentially wiser to go for a higher-level language depending on your character.
But in the end, it's up to you. I guess the fact you've already got a Python book may be somewhat of a deciding factor !