One of Armenia's most popular monasteries, Haghartsin is nestled in lush forested mountains about a half hour drive from Dilijan. The monastery is in good condition and has a resident priest who sometimes gives tours. There are a few different churches, a large gavit, a huge dining hall, khachkars and a massive hollow tree. A slightly sweet bread is often being baked in an old oven.
Goshavank monastery is surrounded by a traditional village, and has a groundskeeper who opens the main church building up for visitors, and can give a tour. This is the one church in all of Armenia where entrance to one of the chambers is not free. The chamber is nice, but you can certainly enjoy the rest of the complex without it, especially since it may not be offered in English. There are excellent carved details and khachkars here, but the "lacework" khachkar at Goshavank is one of the most impressive in the world.
Matosavank is a 25 minute hike from the Dilijan Mineral Water plant. It is easy to miss, and not terribly interesting from the outside, but the inside is a cool, moist, green, mossy world unto itself. The low light illuminating bright green algae covered khachkars is very soothing, and the environment transports you far from your daily life.