Should a first time veteran homeowner try to pay off his mortgage off as soon as possible? What is the Pro and Con of it?

Sgt Davis Le Good question with no simple answer. There are plenty of sources you can read to help make this decision. In my opinion, if you have a low interest loan and can use mortgage interest on your tax return, it might be better to keep paying it. If it is a high interest rate, it might be better to pay if off early. Having debt is not a good idea unless it is a mortgage at a low interest rate.

https://money.usnews.com/money/blogs/the-smarter-mutual-fund-investor/2015/12/03/6-things-to-consider-before-paying-off-a-mortgage-early

https://www.cnbc.com/2017/06/22/pros-and-cons-of-paying-off-mortgage-before-retirement.html

Finance the entire loan for the longest period you can. Knocking the payment down, then double up on the payments. Each month the second payment is paying 100% principal.

I'll add that if the interest rate is low, consider taking the delta you'd pay and investing it in stuff that has a reliable 2+% more rate of return. That way you get the write off at tax time and better net earnings. At some point the lines get closer to intersection depending on your income level. I found it much better to keep more in my IRA kitty given I got a great mortgage rate. So mortgage tracks at 3.5% and the investments have been running at 7%. I'm already paying double on the house to make it a 15 year mortgage and will likely kill that off NLT year 9 when the deduction doesn't make any more sense.