Across Armenia, you can find bed and breakfasts that are pleasant and will give you a true taste of Armenian culture. The language barrier will be significant in the rural areas of Armenia if you do not speak Armenian or Russian, but if you take a phrase dictionary with you, you should have no trouble, as people are patient. If you don't personally know any Armenians, one way to access the true Armenia, away from the Westernized hotels and "Armenian branded" hotels is to find a reliable travel agent based in Armenia.
In Yerevan, there are a couple of hostels. Outside Yerevan, there are a few main recreational areas that offer very reasonable accommodations, but you will be required to live without some conveniences. At the high end are some hotels on Lake Sevan and in Northern Lori Marz 50 kilometers from the Georgian border. Here you will miss nothing, but you will pay Western prices for the accommodations. Around Lake Sevan, there are numerous types of cottages and hotels. Prices are reasonable and start at about $10 per day for a cottage with electricity and within walking distance from Lake Sevan. The city of Sevan, due to its proximity to Yerevan, is the most popular place on Lake Sevan but the history, culture and non-Western feel of the accommodations change as you go south on Lake Sevan.
Tavush Marz is a wonderful place to summer. Dilijan and Ijevan are wonderful towns in which to be based, with day trips to the many ancient churches that pepper this remote region. Costs are very reasonable and Dilijan is known for its sanatoriums from the Soviet era. Do not expect hot water all hours of the day, but you can have a lovely room that will accommodate a family, including food for about $20 a day. Take another $20 to hire a car for the day to visit the surrounding historical sites.
Lori Marz is the second most beautiful region after Vayots Dzor. It has many health resort areas such as Stepanavan, Dendropark Sojut next to village Gyulagarak. Lori is considered to be the Armenian Switzerland. It has numerous churches, monasteries, medieval bridges and monuments. The Stepanavan area is great for hiking, tasting fresh diary products, etc. Small hotels and B&Bs are available in the area of Stepanavan, Odzun, Tumanian, etc.
Tzaghkadzor is a well-known winter retreat. It has many lovely hotels and is popular year round. Check with a travel agent to find the best deal depending on what activity you are looking to undertake. Jermuk, made famous by the bottled water of the same name, is a wonderful get away, but will again require you to leave your western expectations behind.