Families will find Armenia very welcoming of kids, from restaurants to entertainment choices. Very little kids may get lots of attention from restaurant staff and even other patrons, so be ready for this. They may hold your kids hand and take them for a little walk without necessarily asking. This is quite common and it is unheard of for kids to be kidnapped in Armenia, so you can enjoy the free babysitting while just keeping an eye on them. That said, Armenian children are usually pretty well behaved, you won't hear them screaming or crying, so you might get some funny looks if yours have frequent temper tantrums.
There are cafes and restaurants with little playgrounds, themed kids restaurants all over, and rides and parks for them at the Opera, by the Hrazdan River, and up at Monument.
For accommodations for a family, renting an apartment could save money and be more comfortable. Staying in central Yerevan will make it easier to get around without a taxi, especially if you are more than 4 people.
To travel outside of Yerevan, there are day tours, but you can often rent a minivan with driver for $100 a day or less, with gas. It just depends how far you're going.