HDMI over CAT and Fibre HDMI (preferably Hybrid Fibre) are both options for you to consider.
HDMI over CAT can be achieved using solid core, non-CCA, CAT6 to each location and in many systems the central 'HUB' will deliver power, video, audio and IR return over a single CAT cable to a PoE Receiver unit in each Zone.
HDMI over Hybrid Fibre is achieved by using a suitable multi-output Distribution Amp plus a long HDMI cable to each Zone.
Conduit is ideal for CAT or Hybrid Fibre if there is any chance of you potentially damaging an installed cable whilst other works are ongoing.
Empty conduit with a an internal pull cord is ideal running alongside an installed cable as a backup should you ever have to pull in new cables.
Whichever way you go plan for the day when the installed cable fails, is damaged or becomes obsolete.
HDMI over CAT can be HD or UHD - the UHD options currently have to use some form of 'visually lossless' compression, there is a new generation of kit on the way which will offer compression free UHD.
Hybrid Fibre is compression free.
As others point out the Source device and connected Display devices plus an AVR or Soundbar you have in the system all play a part on the reliability of the system.