Centos 6.* comes with Python 2.6, but we can't just replace it with v2.7 because it's used by the OS internally (apparently) so you will need to install v2.7 (or 3.x, for that matter) along with it. Fortunately, CentOS made this quite painless with their Software Collections Repository
sudo yum update # update yum
sudo yum install centos-release-scl # install SCL
sudo yum install python27 # install Python 2.7
To use it, you essentially spawn another shell (or script) while enabling the newer version of Python:
scl enable python27 bash
To install additional libraries, you will need to install PIP:
cd /opt/rh/python27/root/usr/bin/ # cd to the directory where SCL installs python
sudo LD_LIBRARY_PATH=$LD_LIBRARY_PATH ./easy_install-2.7 pip
once installed, you can install PIP using pip2.7, e.g.:
sudo LD_LIBRARY_PATH=$LD_LIBRARY_PATH ./pip2.7 install requests
NOTE: if your username doesn't require root to install software, then LD_LIBRARY_PATH and PATH is set up for you automatically by scl. Also keep in mind that using SCL outside a shell (e.g., cronjobs) isn't quite straightforward. Also, using virtualenv poses a challenge as well.