No, we also use 'trainen'. Sometimes we use 'bewegen' as well: 'Bewegen is goed voor je', but better stick with 'sporten' ('sporten is goed voor je'), because 'bewegen' doesn't always work.
When playing a sport we say 'spelen'. 'Ik speel basketbal.' 'Jij speelt voetbal.' There's a different way to say this as well: 'Ik zit op basketbal.' 'Jij zit op voetbal.' Note that when you are saying the latter, you are not playing that sport right now. Also, 'spelen' only works with games. If the sport isn't a game, e.g. gymnastics, use 'doen aan'. 'Ik doe aan turnen.' 'Jij doet aan atletiek.'
Exercise can be used like that, but its most common meaning is physical activity, which is what it means in this context. But does sporten include the whole range of physical activity that in English would not be sport but still be exercise? For instance running, (non competitively), dance, yoga etc.
"sporten" is vaguely defined in the dutch language, for me it is everything that makes you sweat, so including running (Definitely) and dance (rock and roll, jive), but so much ball room dancing, not yoga. And chess, checkers, playing cards are for me are no real sports but some say they are.
@Hazelbric: Really? Is dance not a sport in English, and is that the case in all English-speaking countries? If people think that dance is not a sport, they have never danced, and certainly not the physically demanding dance disciplines, in which the dancers have been worn sports shoes to prevent injuries for decades now!
In American English we just say we exercise (rather than take exercise). We might also say "I run for exercise" or " I do yoga as exercise".
Overall I'm sad that I talked about "sporten" with Dutch friends for months and thought it meant "to play sports", not just to exercise :( It's a tricky word.