The trailer opens in a desert with lighting storms in the background. This footage was shot in Jordan and from the first frame it's clear that Discovery is cinematic. It's generally agreed that the footage in this trailer is all from the pilot. The Director of Photography on the pilot was Guillermo Navarro (Pan's Labyrinth, Pacific Rim). Some are speculating that this desert is Vulcan's Forge. That's certainly possible. The storms make it difficult to tell whether the sky is red or not. In these shots we are introduced to Michelle Yeoh as Captain Georgiou and Sonequa Martin-Green as Michael Burnham.
Next, we hear a male voice saying "Great unifiers are few and far between but they do come. Often such leaders will need a profound cause for their followers to rally around." It turns out this is Sarek, or rather a hologram of Sarek, played by James Frain. It seems as though he is addressing Burnham which makes sense since it is rumoured the series will deal at least partially with Burnham's rise to command. It sounds as though Sarek is giving Burnham advice. Is he a mentor of sorts to Burnham?
We next see Sarek, still in holographic form, saying (presumably to Burnham) "What have you done out there on the edge of Federation space?". It seems as though the two are communicating over some distance and the question is informal enough that it suggests Sarek and Burnham have some significant relationship. The idea that Sarek and Burnham have a relationship is further supported by the fact that the hologram seems to be in crew quarters which would make it a personal communique from Sarek to Burnham.
The next shot of Sarek is him in-person rather than a hologram but it's suggested that this is a flashback because the following shot is a young person, presumably Burnham as a child. Sarek says, "You will never learn Vulcan. Your tongue is too Human." The child's haircut seems to be in the traditional Vulcan style which suggests we are on the planet Vulcan for this flashback.
Perhaps Burnham was raised or educated on Vulcan? Was Sarek Burnham's teacher or guardian? if an orphaned Human was left on Vulcan it might make sense that Sarek would adopt that child since he has experience with a Human wife and with raising a half-Human child (Spock).
Sarek's costume, to me, is reminiscent of Michael Kaplan's Kelvin Universe Vulcan costumes. The high collar reminds me of Spock's learning pod costume or of Spock Prime's parka. iMDB currently lists some costume assistants but not the costume designer so I'm not sure whose work we are seeing.
That's pretty much all we get in the trailer that might be related to Vulcans. Some additional details and information will probably trickle out over the summer but we'll have to wait for the series to premiere in the Fall to, ahem, discover the rest. i'm looking very forward to the new show!
CBS released the below poster which seems to feature Michael Burnham giving the Vulcan salute. That's further evidence that Burnham might have close ties with Vulcan.