As I know there is the one common language group, Finno-Ugrian, but I'm not sure if that means it is possible to go to Finland and speak to Finnish speakers with my Udmurt language knowledge.
No, it will not be possible. Most Finno-Ugric languages are so different, that they are not mutually intelligible. Even if you speak Udmurt, you will not understand Finnish and Finnish speakers will not understand you.
There are plenty of online resources for hearing the Finnish language if you still want to give it a try. Just search e.g. for any arbitrary Finnish YouTube video.