Buses depart regularly from almost in front of the Covered Market (Pag Shuka) on Mashdots. Fares are under a dollar and take you straight to Echmiadzin past Zvartnots ruins (before reaching town), S. Hripsime (right at the start of the town) and continues straight to the town circle, which is right next to the Echmiadzin compound. There is a statue of Komitas by the town circle as well.
Bus No. 111 and 203 goes straight to Echmiadzin. One of the bus stops is on the corner of Mashtots Ave and Lusavorich St (for bus 203, a bit before the covered market, where the bus stop sign is (many city bus passing there, so a lot of people waiting)) , the fare is 220 Dram (300 dram for bus 203 as of april 2019). It takes less than one hour. There are also marshrutkas depart from central bus station which costs 500 Dram.
Much of the town that tourists visit is accessible by foot. Taxis are also quite cheap, and a couple of Marshutni lines run through the town.
The Echmiadzin Compound has a bookstore/souvenir store inside, with a collection of religious gifts, icons, books, dvds, crosses, jewelry, art, crafts, shirts, and other things. Worth popping in for a look.
Echmiadzin is famous for its kufte. If you've never had "Echmiadzin Kufte" before, they basically take a hunk of beef, then they grind, beat and whirl it into the finest paste, roll it into a big ball, and boil it. Sound appetizing? Well then they throw a hunk of butter on it to melt, and give it some flavor. Worth trying, while you're in town. Go to the town "shuka" (market) and ask for it, there's a few places that specialize in it around.