Python 2.7 is the last release of 2.X and will be mostly unsupported in the future (But not the near future). Whereas Python 3.X is where the latest updates for Python will be happening.
Personally if you are just starting to learn Python I would say go for 3.X since that is the future for the Python language.
Though which version you should go with really depends on what you want to get done/learn. Some modules haven't made the jump to 3.X since it is quite a major undertaking moving large user bases to a non backwards compatible release and they want to give their users plenty of time to prepare for it.
My advise is if you are looking to learn/do a specific thing in Python check and see if it has good 3.X support and then go from there.
For example if you are looking to learn Python to do Dynamic web applications you should be fine using 3.X since the major library for that (Django) has been ported to 3.X.
But if you are looking to use a Module like Twisted (Networking module) you are probably better off using 2.X since it hasn't been fully ported to 3.X. Though don't plan to stick with 2.X forever since almost every module for Python is either ported to 3.X or working on porting to 3.X and soon or later they will drop support for 2.X.
There are a few more things that you could consider when choosing 2.X or 3.X but I don't think they would matter much to someone just learning the language.
But anyways the short answer is if I were you I would go with 3.X since that is where the future of the Python language is and all the main Modules you will likely want (Like Django, Numpy, ect.) have good 3.X versions. Hoped this helped a bit and if you want more information you could try looking at this site which talks about this subject more in depth https://wiki.python.org/moin/Python2orPython3 .
Anyways best of luck with Python and if you need any help while learning I would be glad to help with anything I am able to.