We'll have a car for 3 days and are planning to make 3 trips outside Yerevan:
This is how I usually organize things for people:
You can do:
Hor Virap – Azat Reservoir – Geghard – Garni – Symphony of Stones – Charents Arch
OR
Hor Virap – Surenavan (the stork village) – Areni (have lunch there) – Noravank – Areni Cave or Magellan Cave – Azat Reservoir, and optionally you can stop at any of the many wineries along the route.
Most people combine Sevan with Garni and Geghard (instead of Hor Virap), but I personally prefer the route Sevan – Saghmosavank – Echmiadzin – Zvartnots, or Sevan – Bjni Fortress – Saghmosavank. That way you get to see some really beautiful canyons.
On the third day, you could go to Noravank if you combined Hor Virap with Garni and didn't continue farther south.
OR
You can head farther south toward the Goris area. There you'll find the Tatev cable car, the cave settlements of Old Khot, Goris, and Khndzoresk, the Shaki and Jermuk waterfalls, and the ancient megaliths of Karahunj. (Though fitting all of that into a single day comfortably would be difficult, so it's better to choose whichever places appeal to you most from the list, or spend a night in Goris.)
These are probably the main destinations, although there are plenty of other interesting places as well.
I recommend visiting Levon's Cave. One man spent 23 years digging it out with nothing but a hammer and a chisel.