The first thing you'll see when you go to the Arena is a list of dueling arenas, or servers.
Battle records will not be saved in this arena. For learning and trying out new cards. If you want to duel your friend, you have to select the Training Arena and create a Duel Room for you friend to join you.
Single Player Arena
This is another Training Arena, except you can duel computer controlled opponents.
Here you can win new cards and gain Experience and Fight money. If you select one only open to levels 1-3 or 1-14, you will be safe from encountering high level opponents. You will be paired randomly with opponents in these arenas.