Most people train in the martial arts for one of two reasons. Most styles of martial arts can provide both benefits, at least to a degree. And well they should. If they don't, I suggest you continue looking. However, many styles and systems differ widely in their approach to both fitness and self defense.
Many schools teach systems that are tailored for competition.
The techniques they teach are more suited for tournament competition and may not be as appropriate for self defense. Typically, you'll find a strong emphasis on aerobic training in these schools. They tend to spend considerable more time developing strong defensive and offensive skill and my not place as much importance of rigorous physical training. Then, there are those rare schools that place nearly equal emphasis on both aspects. Attend class as a guest or visitor.