It depends how old you are, and if you want to put the shows in the proper context, not just in the Trek timeline, but in actual history. I urge you to watch TOS, but maybe watch the top 10 episodes, that are generally considered the best. City on the Edge, Space Seed (ties right into Star Trek II: The Wrath of Khan), Doomsday Machine, Trouble with Tribbles and maybe a good 'Federation episode' like Errand of Mercy or The Enterprise Incident. Don't try to sit and watch all 79 right away as you will quickly realize any series can be hit or miss and might take you out of a greater undersanding of what Trek was.
If you like these few old episodes, then by all means, watch them all, they all have something that we die-hard fans love, or love to hate.
Then move on to TNG, I don't think you should skip a whole series just because some love others more. Sample TNG, the pilot Encounter at Farpoint, then skip to some of the better shows. Big Goodby, The Neutral Zone (for continuity's sake), Best of Both Worlds, Family, Inner Light etc. etc.
Do the same for the others, remembering that DS9 started towards the end of TNG run, so there is some great cross-over and tie ins that enrich the trek universe and enjoy it, for your own personal reasons.