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.
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.
Cathedral of Echmiadzin. Founded in 301 by St. Gregory the Illuminator and rebuilt and added to for over 1,700 years, Echmiadzin now is the heart of the Armenian Church, and seat of the Catholicos of all Armenians. The altar is built over an ancient pagan fire worshiping pit (accessible from the museum behind the altar), the inside has some interesting frescoes, altar, lamps, and thrones. The intricate bell tower entrance is an addition from the 1800s, and the surrounding grounds have gardens with examples of khachkars from throughout Armenia and further. Near the new entrance gates are a few exquisite khachkar examples from Jugha (now in Azerbaijan) - which had by far the largest khachkar collection in the world. The thousands of khachkars which remained there were destroyed during this decade by the Azeri government.
S. Hripsime Cathedral. An outstanding example of earlier Christian architecture, this 7th century cathedral with soaring ceilings and vast interior space was an example for future architects.
S. Gayane Church. Built in honor of one of the Christian virgins killed by the Armenian King, along with Hripsime, this smaller church with an arcade in front is a few blocks south of the Echmiadzin compound.
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.