English

1. I am into sports.
(What is the meaning of this sentence?)
This means, "I am very interested in sports" or "I am very involved in sports."

2. Exercise is good for the health.
2-1. Exercise is good for health.
2-2. Exercise is good for your health.
2-3. Exercise does good for people.
(Are all correct?)
2 is incorrect; all the others are correct.

3. Our athletic meet took place yesterday.

(is this correct? Do you have better expressions?)
A "meet" is usually used for track and field or cross country competitions. All the rest are referred to as games or competitions. "The school's football playoff game will be held in another city."

4. Moderate exercise stimulates the blood circulation.
4-1. Moderate exercise stimulates blood circulation.
(Which one is right? Do we have to use 'the' or not?)
4-1 is better than 4. You can use "the" but it sounds clumsy.

5. I like light exercise like jogging.
5-1. I like heavy exercise like weight-lifting.
(Are they both grammatical? Do we have to put 'a' before 'light exercise' or 'heavy exercise'?)
Both are correct; you can use "light exercise like" or "heavy exercise like" if you wish -- or not.

6. Exercise can cut down on stress.
(Is this right? What about "without 'on'?)
6-1. Exercise can cut down stress.
6-2. Exercise can reduce stress.
(Are both grammatical?)
All are grammatically correct, but 6 and 6-2 read the most smoothly.