The most common worship service that Byzantine Catholics offer is the Divine Liturgy (what Roman Catholics call "mass"). Much like our Western (Roman) Catholic brethren, the order of the service is largely the same, but some of the prayers differ, as do the readings. In addition, in the Eastern church the liturgy is largely chanted, versus the spoken prayers of the Western Church.
However there are many services that parishes may offer. This link to the Metropolitan Cantor Institute explains the daily cycle of services, complete with links to some of those services!