No, not quite. I'll take the standard 'quarter note' (I call 'em crochets, but I'm old, so...), and count, as you said, as '1 2 3 4 ...'. For eighth notes, I'd count '1 and 2 and 3 and 4 and ...'. Sixteenths become a bit more tricky, but the principle is the same, with 1 uh and uh 2 uh and uh 3 uh and uh 4 uh and uh ...'. Try it out at 60 bpm (so 1 2 3 4 at 1 second intervals...). Repeat, at the same tempo, still with the 1 2 3 4 at second intervals, but with the 'and' interjection. Once that's done, do the same with sixteenths (you may want to slow the metronome down a little at first, though; it's not that simple for the tongue and respiration...).
There, that's all there is to that. Next week: triplets and paradiddles...