Contemporary usually means music. It means likely a lack of, or total abandonment of liturgy, and lots of loud music.
Done poorly: A band plays the latest Christian hits you hear on the radio while you try to follow along with the words on a projector screen, the book of worship is absent and liturgies are thrown away. (To add to the stereotype, the Pastor is probably in blue jeans, and has a soul patch).
Done excellently: A band plays modern music and hymns that are easy to follow along, maybe one modern Christian song as 'special music', and plays/sings in an 'accompaniment' format, instead of a 'concert' format, understanding that worship is about WORSHIP, not entertainment. (But even the old hymns were new once!). Liturgies are present but perhaps they'll drop certain age-old traditions like reciting creeds or using the old orders of worship. Liturgies are certainly used for communion, etc. You may still see the Pastor wearing a robe/alb and stole (not that I'm implying the robe/alb and stole makes 'excellent worship', just that you'd never see it in the 'bad contemporary' because there's a huge focus on trying to make it not look like church).
Traditional, again, in the UMC usually has to do with music. Old hymns, from the United Methodist Hymnal, or older books like the Cokesbury Worship Book.
Done poorly: "We've always done it that way". They use old liturgies and traditions reciting boringly and with no enthusiasm because that's the way it's always been. They don't have any theological or intentional reason behind it. The hymns are old and they'll never break out anything new because it's not what we've done before. They'll recite creeds that, despite reciting them thousands of times they haven't memorized because they are just boring-ly reciting them and not intentionally reading them and making any sort of oath. (But they won't stop doing it). The Pastor is definitely vested!
Done excellently: Liturgies appropriate for the season are used not as a sign of the way things always have been; but as a sign of ecumenicalism, heritage and structure. Older hymns are used and sung passionately, and new hymns are often introduced (whether old or new; but not necessarily 'contemporary' new. New hymns are written all the time). Vestments, dress and paraments used in the sanctuary and by the Pastor and whomever else is a part of worship are intentional and 'there for a reason', not just there because they've always been there. In the best examples of this, you could ask someone why a particular parament or decoration is there and they can tell you what it means instead of just telling you who donated it.
So, IMHO, there are 4 types of worship common in the UMC. Good traditional, bad traditional, good contemporary, and bad contemporary. You'll kind of have to visit the church to see which they have, and see what fits with you.
There's also a component called 'high church' and 'low church' that can be in contemporary or traditional. TEC is almost always 'high church', whereas most UMC congregations are 'low church'. Meaning that while we use liturgies, we don't follow a rigorous liturgical worship format that TEC does. You likely won't see any chasubles (if any vestments), a processional at the beginning may or may not happen, communion is not usually done at every service, and while there are readings and liturgies; even in a traditional worship setting it won't be as liturgy focused as TEC worship. We DO have high church worship UMC's, just not many.
As a musician myself I may be a tad hypercritical.
