I would say, pick the ten best episodes of TOS by fan consensus and watch them. Then watch a lot of TNG and DS9. If that doesn't make you a Trek fan nothing will.
Why? Why should someone bypass TOS for TNG/DS9 when they were seemingly drawn in by the big-screen action-adventures of Kirk and Spock. That would be like me telling someone who liked First Contact
and an episode or two of TNG to go watch TOS instead.
Possibly because people drawn in by the NuTrek may be turned off by the comparatively slow pace of TOS.