Looks like the seedling tray system is working well, I've already got a few sprouts popping up!
They're a few days early too, they're supposed to take 7-10 days to germinate but I saw the first signs of sprouting only 4 days after I started them. I guess that's a good sign that I'm doing something right!
As an aside, I wanted to check the exact phrasing for the quote in the title of the post, only to discover that it's not actually a quote from Star Trek itself, but instead from a parody song Star Trekkin'. Who knew?