Hehe, why don't you go to both!!
Vespers is a lovely evening prayer service. If you are thinking about attending a Sat night Vespers, then timing wise, it is the beginning of the Sunday services. You'll see and hear prayers and liturgical action that recalls the creation of the world, the fall, and the promise of the Savior--basically the history of the world up until Christ. I'll post an article for you on this later. It typically takes about 45 minutes.
Divine Liturgy is the center of the Orthodox faith. It is the Eucharistic service. I think you'd find some similarities between the RC Mass and the EOC Liturgy. I will also post a resource for you on this which, if read carefully in advance, would hopefully help you understand what is going on and when. Either service would be fine to attend!
Hints for visiting: Do what other ppl do. At my church, the building is so small that there's no side aisles. When the priest processes around the church, censing the icons and walls, we have to shuffle to get out of his way. Some churches may have pews and so people will stand or sit at certain times. You can cross yourself if you wish, and a good rule of thumb is at every mention of the Holy Trinity plus whenever you want to. But you in no way should feel obligated to do anything worship-wise. You don't have to venerate the icons, go up and receive a blessing from the priest (since I'm sure you're well aware that Orthodoxy reserves the Eucharist for those who are prepared to receive it, who must be baptized Orthodox Christians) or feel like you need to sing along with the choir.
Just go, relax, observe, and enjoy!