Typical ceremony song choices are as follows, either instrumental or vocal renditions.:
Groom: Mi Bon Siach, Mi Adir (from the Siddur marriage section)
Bride: Eshet Chayil, Ani Ledodi, Boi Beshalom, etc, or a folklore tune e.g. Erev Shel Shoshanim
Other more popular modern tunes are also appropriate for either, eg. Fiddler on the roof melodies (e.g. Sunrise, Sunset instrumental), or Tum Balalaika in Yiddish, folk songs of modern Israel and so on.