Renting in Armenia
What to look for
As the infrastructure of Armenia improves, natural gas lines, 24 hour water, and digital phone lines are becoming more and more the norm. In the meantime, when you are searching for a place to rent, these can be important considerations. The gas is especially useful for heating your place cheaply in the winter. The digital phone lines make using the internet (among other things) much easier, and are already available in much of Yerevan.
The best choices are either a "Kari Shenk" (stone building) up to 6 stories high, or a private home. Which you choose is a matter of personal preference and what is available. The very tall buildings (over 6 stories) are usually made of cement panels and are of dubious seismic safety.
How to search
Classified ads are easy to find in Yerevan, I like Gint, but these are usually in Armenian or Russian. Brokers are another way to go, they usually charge their finders fee to the apartment owner. In both Yerevan and outside, word of mouth and asking around are a good way to find a place.
Yerevan or not
If you are considering moving to Armenia and are not sure where you want to live, Yerevan is the monolithic center of the universe. There is a bit of cultural life in towns outside, some "night life", shopping, services, but these are all centered in Yerevan and much easier there. If you want the countryside, peace and quiet, dirt cheap living, etc, then you can live anywhere in the country. With the new roads, getting to Yerevan is not difficult, and you can always locate an hour from Yerevan so that a visit is easily manageable.
It is almost silly to discuss rental prices "in Armenia", but I will try. In the extremely central parts of Yerevan, a small place that is not remodeled is about $200 a month now. A nice one bedroom may be $500. A fancy house may cost $2,000 a month. If you want to rent only for a few weeks or a month or two in the tourist high season (summer/fall), the price can double. A metro stop or more out (Paregamutyun, Sasuntsi Tavit, etc) will see a significant drop in price, around 50%. Then the further the cheaper.
In the "regions" of Armenia prices are drastically lower than Yerevan. Even cities like Gyumri and Vanadzor are no exception. Towns like Dilijan, Ejmiatsin and Goris are even cheaper of course, and villages are practically free. It doesn't make sense to rent in the villages if you are going to stay very long, since you can just buy a place (if the land purchase is an issue you can get a 99 year lease on it).