Should I start investing now or should I start only after I have fully paid off my student loans?

Would suggest clearing off your student loans first so as to minimise the interest payable

Do both, but a bit more on clearing debts. This is because, interest payable on loan is guaranteed, but investment returns are not.

Also, clearing the debts relieves you from the burden of worrying about it.

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Note: do not go into more debt for investments. It does not bode well, coming from someone who tried to leverage for stock investments.