Is there continuity I would miss if I skipped X-COM and went to X-COM TFTD?

They're very similar games, only the difficulty is much higher on X-Com 2. If you're interested in the plot that is revealed through researching things, you will miss some things by skipping right to number 2, so I would definitely suggest starting with the original.

I originally played Terror From the Deep before playing the original X-COM, and I had very little trouble following the plot. Though they are related (some of the aliens from the first make an appearance in the second), you will be just fine playing the second first if you want to. There are some minor UI / Organization changes that make the second a little easier to manage.

I've played the first 3. While there is some plot that carries over between 1 and 2 you're not missing anything in doing them out of order. Psionics is greatly nerfed in 2--in 1 if you develop your psionic characters enough it becomes a cakewalk. (For a challenge I went into the final battle with no weapons at all. I was able to win the first battle without ever firing a weapon, I had to fire ONE shot in the second.)