Re: GYM
In base game, there are 2 premade gyms that come with the game. One in Oasis Springs and one in Willow Creek. There are also gyms in Windenberg (Get Together), Del Sol Valley (Get Famous), and San Myshuno (City Living) that are premade gyms. None of the vacation worlds have a gym but you do have the ability to put a gym on any lot in all of the main worlds if you don't have one and want one.
So all you have to do is travel to one of those and exercise on a weight machine, treadmill or boxing dummy.
You said you have tried to travel with no luck. Are you having some type of issue that won't let you travel or something?
Hope this helps.
