Advice needed- already strategizing about my second property...

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I'm buying my first property (finally!) with a FHA loan, it's a BEAUTIFUL three unit house in a good neighborhood with a strong rental market, meets all my requirements (off street parking, good schools, quiet street, good aesthetics, etc,) and it's fully updated- needs nothing! This house isn't going to be a huge money maker right now because I will be occupying one of the units, but I will come out a little ahead and be able to live for free so I can start building equity and the money I save from not having to pay rent or a mortgage will allow me to start saving for my next property.

So my question is- whats the best strategy to acquire my second property? Should I sink extra money on the mortgage for now, refinance to a lower rate and use a home equity loan to as a down payment on the next house? Should I just keep the mortgage as is and start banking every penny I have into a savings account? With my current job and 'living for free', I should be able to save around $1-1.2k a month and if I get a second job I could potentially save bank another $1k. 

I really don't want to use a third party to borrow the down payment from- the less people I owe money to the more comfortable I will be, I would much rather use my own cash/equity.

Any advice is appreciated! Thank you!