On your adventure, even when surrounded by Special Events/side-quests and all that, your main goal is to travel through all the regions and earn your eight badges.
Then you will go on to the Elite Four and hopefully defeat them. When you are named Champion, there are still more things you can do. (Don't get your hopes up, though - it's a long way ;))
Gym battles are like any other battle. A moderator will take the role of the gym leader. If one moderator is absent for a long time, another moderator can feel free to step in if the other is offline.
Please don't get frustrated if you lose on your first try - gym battles are not supposed to be easy. ^-^;
When you defeat a gymleader, you will be given a small badge to put in your signature (if you wish) and your Trainer Card (required).
Traveling
Make sure you go through all the routes/cities and the regions in order. You will stumble across a gym or such from time to time, or a special event, etc. You may not go backwards. So, if you're done with Route 1 and move on to Route 2, you will not normally be allowed to go back to Route 1.
In each route/city, there will be a pinned topic stating what the route/city looks like, what sorts of Pokemon you'll frequently see, etc. Your adventure will be easier if you read what things there are in the route/city. They're especially useful for telling you what's in the city and what events to look out for.
If you are done with one route and want to move on to another, please state in a final post on your adventure thread that you are done and will move on. Your thread will then be moved to a new area so you may continue there.