HDMI? HD-SDI, DVI?
All the above and then some?
HDMI, DVI, DIsplayPort, etc. over UTP/STP/CAT solutions may be one option if you are dealing almost exlusively with a single source to a single destination. But if you start looking at sending one input to multipel destinations or swithcing between destinations then it can get more complex due to having to address potentially different supported resolutions, EDID, HDCP, etc.
HD-SDI may be the simplest in terms of patching and distribution but part of that is because it does not deal with HDCP or EDID. However, that also means not being able to support encrypted content and potentially having to do more manual configuration.
DVI and DisplayPort have the advantage of potentially handling both digital and analog signal formats, but you may have to deal with EDID and HDCP. HDBaseT or AVB based system may also offer an option for handling multiple digital and/or analog signal formats, but you are then also into more complex devices and routing.
What it really comes down to is having to have a good understanding of the potential usage and your goals in order to make any specific recommendations.