The best graphics card you can get for the $150 price range is the Radeon R7 370. It is a little faster than the R7 360 and you can get a good R7 370 like this one for $150 http://amzn.to/1O1PYqX
That Gigabyte one is really well reviewed but if you have other brands you like feel free to research them and see how they compare.
You will need a 500w minimum power supply for the R7 370. I looked up an R7 360 and it also said 500w or higher.
If you upgrade to an R7 360 or 370 the new bottleneck will be the CPU. From there you would just need to save for a Phenom II X4. The fastest one your motherboard supports.
Side Note: How high you can run play also depends on the specific game. An example is, a computer may be able to run The Sims 3 maxed out, but only be able to play The Witcher 3 at low graphic settings.