It depends on the activity. (These are usage examples--I don't do all these things!)
Sports: "play."
I play golf. I play tennis. I play chess. I play video games.
Exercise: just use the verb "to exercise," or use "take."
I take exercise. (But that's a little old-fashioned or perhaps British. I personally DON't say that).
I exercise.
I exercise twice a day.
I exercise at the gym.
Special use: DO exerciseS meaning a specific exercise.
My physical therapist has me do exercises for my knee.
To maintain my fitness, I do exercises like push-ups and sit-ups.
To do a SET of exercises for fitness and muscle tone is called "working out," or "a workout."
He looks like he's in good shape, I bet he works out.
I work out every day. On a business trip, I always choose a hotel with an exercise room.
For activity in general, if the activity can be used as a verb, just use the verb.
I run.
I walk.
I go for a run every day.
"How do you like to exercise?" "I walk."
For one specific activity at one specific time, "take."
Would you like to take a walk with me?
I'm going for a walk now, I'll be back in half an hour.
I'm taking a walk to the supermarket, do you need anything?
However--as always, when I start to think about it I realize just how complicated it is--when it is, specifically, an organized group event, you sometimes use the word "do."
My cousin did the Boston marathon this year. OR,
My cousin ran the Boston marathon this year.
She did the Pan-Mass Challenge in 2005. OR,
She rode the Pan-Mass Challenge in 2005.