From MacMillan Dictionary: Take up: to start doing something regularly as a habit, job, or interest Examples: I took up smoking when I was at school. Chris has taken up jogging.
Start up: Meaning: to bring a business, organization, or project into existence Examples: The Agency helps over 1,000 firms start up each year. She left the company last year to start up her own business.
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Take up (PHRASAL VERB): take something ↔ up to become interested in a new activity and to spend time doing it. Example: Roger took painting up for a while, but soon lost interest. She took up tennis at the age of 11. You should take up swimming. Source: Oxford and Longman Dictionary of Contemporary English.