As someone who hasn't played any of the Trails game:
How tight is the connection between Trails in the Sky and Trails of Cold Steel? Do they lead into each other or is it more along the lines of how Final Fantasy is a series? (not between 1-3 but between the two trilogies)
It's nothing at all
like Final Fantasy in that regard.
Trails games are all heavily connected
to one another, and each sub-series (which are: Trails in the Sky, Crossbell, and Trails of Cold Steel...with another additional sub-series coming in the future) directly continues characters' personal stories and also overall plot lines from the previous sub-series(es).
So by the way, that means it's not "two trilogies" as your post says: but rather it's
1. One Trilogy (Trails in the Sky).
2. One Duology (Crossbell, which has two games).
3. Cold Steel series: this is not not a trilogy as it currently has 4 games in the sub-series, not three...but part 4 has not been released in English yet, even though it's been out in Japan since 2018. Moreover, they are releasing a 5th game this year in Japan that is going to continue the characters and close up the loose ends from Cold Steel, and also the continue the characters and close up the loose ends from Crossbell too...and also add new characters that sets up the next upcoming sub-series).
Some of the plot lines that start in the Trails in the Sky series, are resumed in the Crossbell series, and then are concluded in the Cold Steel series.
Also, for these reasons, each sub-series heavily spoils the previous ones, so if the player plays them in a different order than they were released in, they therefore will have a very diminished experience for that reason.
As well, players who skip some of the earlier games will have no idea what the significance of many elements of the world and returning characters are, since they were explained in previous games, but do not get re-explained (or sometimes they receive only a minimal
re-explanation) in the newer games wherein they re-appear.
The best thing to do is to play the games in order of their release that they came out in Japan, if possible.