Not a bad start.
Based on these specs you have a "350 W Power adaptor" and a mid tower case. But you say 300w PSU. Do you know which it is?
http://www.asus.com/Desktop/Entertainment/Essentio_CM1730/#overview
Assume you want a gaming build, not something else.
With a $300 budget you cannot upgrade CPU, MB and Video. So leave the CPU and MB alone and just upgrade video.
Two big options for upgrading video. (1) Keep the 300/350W power supply and buy a video card that fits in that power window or (2) install a new power supply along with your new video.
For me, given the rest of the system, I'd spend $135 on an HD7770 and keep the rest of your money for when you upgrade the CPU and MB in the future. HD7770 (80W, 100W max on OC) will work fine with the X2 220 on a 300W PSU.
You will see a huge improvement over your current gaming with integrated adapter.
If you want to upgrade PSU and then install a new card then a power supply like this Antec $40
http://www.newegg.com/Product/Product.aspx?Item=N82E16817371045&Tpk=antec%20vp450 can be coupled with any card in this article no problem.
http://www.tomshardware.com/reviews/gaming-graphics-card-review,3107.html
If your $300 budget was MUST SPEND money, then look at $40 for power supply, $250 HD 7850 for rest.