Hello there Kevindong, congratulations on your new home!
There are a few things that I would like to mention about your post:
1) Your pump is leaking.. this looks like it is coming from where the pump bolts onto the housing. Could be a bad o-ring, could be a crack, could be anything.
2) If you have a home warranty, it should cover the SWG. I have only heard a few stories about this though, so I am not 100% certain. Also, have the home warranty cover the pump repair--if you do indeed have a home warranty, this definitely SHOULD be covered. Often times this stuff is looked at by a professional before purchase; I wonder why this was not done for you prior to closing the sale.
3) As others have stated, your check valve on your spa line is either not functioning properly or does not exist. If you do not have one, this will continue to happen, and there is not much else you can do to stop it. Installing one is not the most difficult thing in the world, but adding things in-line can be fairly tricky if you've never worked with plumbing before. If you do have one and it just so happens to be see-through (straight swing), it will not be difficult to repair. The clear cover comes off and swaps with a new one.
4) Just a heads up that your pump actually looks like a SuperFlo judging by the visible portion of the sticker and shape of the housing. It could be 1.65 THP as suggested by someone else, but the best way to know for sure is to look at the labels on the black motor.
All in all, I would certainly recommend having home warranty take care of the bigger issues if possible. If it is not possible, you can opt to do it yourself or hire someone to do the repairs. If you do it yourself, I HIGHLY recommend ordering the parts needed online. I am certain that there are topics posted about plumbing and repairs around here, as well as YouTube videos.
By the way, before changing any valves, attempting any repairs, etc, do you have a good understanding of how the system flows? How the valves work? If not, I can draw a diagram on a pic you upload and explain it, or I bet there's a sticky posted somewhere that someone can throw in here.