The Orthodox churches
The Orthodox Church is the Christian church of the East, which separated from Rome in 1054. In the US it is organized by heritage: Greek Orthodox, Antiochian, Russian, Serbian, Romanian, Ukrainian, the Orthodox Church in America and others, all sharing the same faith and worship. Many parishes worship mostly in English. The Oriental Orthodox churches (Coptic, Armenian, Ethiopian, Eritrean) have a similar style.
Entering the church
You will see icons everywhere, and an icon screen, the iconostasis, between the nave and the altar. Orthodox Christians often buy a thin beeswax candle at the entrance, light it and kiss an icon. Visitors may light a candle too, or simply look around. Many US parishes have pews; in others most people stand, with chairs along the walls for anyone who needs them.
Don't worry about arriving late: in many parishes people arrive during the opening part of the Liturgy, or the Matins service before it.
The Divine Liturgy
The Liturgy, usually that of St. John Chrysostom, is almost entirely sung: the priest and the choir or chanter alternate, and the congregation joins in familiar responses such as "Lord, have mercy". Incense is offered throughout. The service includes the Epistle and Gospel readings, a homily, the Creed and the Lord's Prayer, then Holy Communion. People cross themselves often and bow; visitors may stand quietly.
Communion and the blessed bread
Holy Communion is given only to Orthodox Christians who have prepared by fasting and confession. At the end of the service, everyone, visitors included, is invited to come forward to venerate the cross and receive a piece of antidoron, blessed bread that is not Communion. This is a good moment to be greeted by the priest.
Dress and etiquette
Dress modestly. In some parishes women cover their hair with a scarf; spare scarves are often available. Men remove hats. Afterwards there is usually a coffee hour in the church hall, where newcomers are warmly welcomed.