You are welcome Porkins76
To address your concerns:
As you will be aware you get what you pay for and you will not get anything other than an entry level video card for less than $200 unless the card is on special offer, see benchmarks here
Swapping out a standard ATX PSU is not as difficult as you may think, we would walk you through it and provide links to tutorials to help.
From research which lead me to here, your present PSU is not adequate for an R7 360 video card as they require 500 Watts & 33 Amps on the +12V rail, if you take off the side of the computer case you will most likely find that the present PSU is the AcBel brand, I personally would not use such a PSU as a doorstop let alone near my computer.
RE your question relating to the MB and any potential upgrades, you will note in my reply #7 that it says " Example upgrade suggestions: " this was to emphasize the sort of quality and performance that you should be aiming for so as to avoid disappointment, please see my canned text below for considerations that I always mention to members who are considering a GPU upgrade;
Please note that before purchasing an add on video card that the following must be considered;
Is the present PSU (power supply unit) powerful enough or will it also need upgrading.
What type of case do you have, full tower, mid tower or SFF (small form factor) this is important as you must make sure that there is enough room inside the case for the upgrade/s, upgrading a SFF type system can be difficult as they often use propriety components such as smaller and differing shape PSUs and they only have enough room inside the case for a low profile type add on card.
Cannot comment on the particular card at your link other than to say that it having no reviews does not instil confidence in me
An article you may find of some help here