Well, first, congratulations, and don't panic.
The M216 is a respectable 16 channel, 4-subgroup mixer that can indeed be used to effectively support a 16 track tape machine. It may have been an impulsive purchase, and believe me, I've been there, but since you're apparently a recording "Newbie", just remember it might take a little time to come up to speed.
The mixer can do what you want it to, in a basic sense, as you've described, but one of the fundamental things you need to make that happen, is familiarity with the board.
Tascam Parts sells manuals, and they're available by phone order, so just call (323) 727-4840, and order the M200 Series manual, which covers the M216. You'll be a lot better off having the manual, right from the start, and I suggest getting one, asap.
I was wondering if you already have a 16 track tape machine, or if it was a planned purchase?
Ok, there's only 8 TAPE INs, on
the M216, but the other 8 LINE INs, on mixer channels 9-16, could be used for additional tape ins, [9-16 from the reel-16]. You'll need an RCA to 1/4" Mono plug adapter, qty 8.
PGM OUTs are the outputs from your 4 subgroups. Channels are assigned to subgroups by a combination of PAN, and depressing either 1-2 and/or 3-4, the white buttons just above each PAN knob.
SUB IN would likely be your L/R Stereo In.
INSERTs are specialized 1/4" stereo jacks, which have both a SND and RCV in the same jack. INSERT requires a special Y-patch cord, but with INSERT, you can patch an effects device inline into each individual channel.
There's really a lot more detail about the specific features of the M216, which would be lengthy to describe. Your best bet is to get the manual, read it, and try to understand the features. It's when you understand the features of your mixer, that you will be able to understand how to apply the mixer to your application.
But, pls, don't panic. IMO, it was a good purchase, that can be adapted to do what you want it to. Tascam boards such as the M216 are very open architectures, and possibilities are only limited by your imagination.
If someone were to come to me, and exclaim the M216 couldn't support a 16-track reel-to-reel, I'd tell him he had not thought it through enough.