Goris is a large town in southern Armenia, and a confluence point for those travelling between Yerevan, Iran, & Nagorno-Karabakh. If you've gotten as far as Goris, you'll be aware of how awful the roads are. The town itself isn't the most charming, but there are plenty of hostels/guesthouses if you need a good rest.
South towards Tatev, Kapan & Iran
Goris is on the main highway between Yerevan and Iran (M2), though there are two routes going south: a westerly route passing through Tatev, with the easterly one passing through Karahunj. Both of these routes converge before reaching Kapan. The easterly route (via Karahunj) is busy with Iranian truckers who tend to very receptive to hitchhikers. Bear in mind that most Iranians are driving battered trucks that are about 25 years old (embargo, eh). Indeed it's very likely see a few of them broken-down on the side of the road.
If you're coming from Yerevan and want to go straight to Tatev (i.e. you've no need to see Goris), get dropped off at the Goris Entry Monument, which is about 4km before Goris. The road for Tatev is right there.
For Kapan or Iran, you take the highway that runs through the downtown, so just keep walking southwards down Syunik Street until you find a good spot.
North towards Yerevan, Sevan
As mentioned above, hitching a ride with an Iranian truck driver would be your best bet, as they're usually going to Yerevan or beyond. If possible, try to get as far as the Goris Entry Monument (yep - the turn-off for Kapan), as there is a store and large siding there where trucks pull in.
Given the terrible state of the road to Yerevan, a ride in a truck could easily take 6 hours, perhaps even more if they stop for a break.
East towards Nagorno-Karabakh
Hitching from Goris will bring you straight to the Nagorno-Karabakh capital Stepanakert. Walk up to the top of Mashtots Avenue, where it meets the M12 highway, and hitch from there. If you're carrying heavy bags, a taxi from downtown will cost you 500 drams to the drop off point (May 2019). Remember you will have to obtain your visa at the checkpoint on the way (more info available on the NG page - but in short, have the name of a guesthouse ready to write down on the form, even if you don't have a booking).