Running cat 6 close to coax and probably makes the most logistic sense if you are just wiring for internet ready devices like blu ray players, receivers, streaming devices, tvs etc.
It really depends on how you intend on using the cat 6. If you read the Home A/V Distribution forum here you will see that a general recommendation is to run 3-4 cat 6 cables to each room where you might eventually have video and hardwired internet access. Many people only use wireless internet access now but some people (me included) prefer the stability of a hard wired ethernet connection for certain devices. The additional cat 6 cables are pulled for potential distribution of HDMI over cat 6 or HDbaseT and/or keypad integration with whole house audio/video distribution systems where the signal source originates from one point like an equipment closet for distribution to different tvs.