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You first have to do some activities for your health during summer vacation: for
example, swimming or other physical training. Make a routine for doing such an
exercise and try to keep the fixed schedule for it.

Do you think semicolon there is a good choice considering the context or other things?
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No, you cannot use a semicolon there; semicolons separate independent clauses. Considering the context, an m-dash would be a good alternative to the colon.
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an 'm-dash' means this one?: —

Do you think using a dash there is better than starting a new sentence like this? ~~. For example, ...~.
You first have to do some activities for your health during summer vacation: for example, swimming or other physical training.

Well, first, 'for example...training' is only a fragment, so you cannot simply put a period after 'vacation' and capitalize 'for'. All in all and considering the register, an m-dash ( – ) is the best choice.