Good thing you ask, because I searched for it before going to the Caucasus last year and couldn't find much. Fortunately, I have the answer to your question.
I did Tbilisi–Yerevan back and forth by minibus/marshrutka. Minibuses are great, I recommend them. They're quite fast, cheap, and it's how the locals do it. Plus, the scenery between the two cities, with its mountains and valleys and green plains, is quite beautiful.
There are minibuses for Yerevan at the Ortach'ala station1 (ორთაჭალა), you can just show up there at a reasonable hour (early afternoon or earlier, I think), and minibuses leave usually when they're full or within ~2h (your mileage may vary). In my case, I showed up at 9:50, and we left at 11:25.
(As to how I found it, what I did was simply to ask a taxi driver for "a marshrutka to Yerevan", and they drove me to Ortach'ala.)
On my way back from Yerevan, they dropped me at Avlabari, so there might be departures from there also.
1 Ortach'ala bus station: 1 Dimitri Gulia St., Tbilisi (coordinates: 41.6753 N, 44.8347 E)