I don't believe it is worth upgrading the CPU since the performance you get will not be justified by the price you are going to pay. I recommend saving up money for a new mobo and CPU. The motherboard only supports socket AM3 CPUs. The desktop was reviewed in Apr 2011. AMD did not release their 1st AM3+ FX CPU until Oct 2011. The first gen Bulldozer FX CPUs were at best considered a "side-grade", not an upgrade based on benchmarks.
The 2nd generation Piledriver FX-8350 was released around Oct 2012 and it is basically AMD's most recent CPU generation. There is the FX-9xxx series which are basically overclocked versions of the FX-83xx series, but they are rated for 220w power as opposed to 125w for the FX-83xx series. You would need a beefy PSU, a good water cooler, and a really good motherboard.
I generally recommend Intel CPUs over AMD CPUs due to their performance. But they are a little bit more expensive and only the more expensive ones will allow you to overclock. AMD CPUs are a bit cheaper, but also less powerful. The FX-8350's performance is generally between a 1st gen and 2nd Intel Core i5 CPU. Intel is currently up to the 6th gen Core series which was recently released back in August; 5th generation Intel Core CPUs were primarily released for laptops, but there are a few desktop models. Below are prices of AMD and Intel CPUs (4th generation Haswell).
AMD FX-8350 = $170
http://www.newegg.com/Product/Product.aspx?Item=N82E16819113284&cm_re=fx-8350-_-19-113-284-_-Product
Intel Core i5-4440 = $190
http://www.newegg.com/Product/Product.aspx?Item=N82E16819116942&cm_re=core_i5-4440-_-19-116-942-_-Product
Intel Core i5-4690k = $240
http://www.newegg.com/Product/Product.aspx?Item=N82E16819117372&cm_re=i5-4690k-_-19-117-372-_-Product
As you can see from the above prices at Newegg.com, Intel Core i5 CPUs are more expensive than the FX-8350. The i5-4440 cannot be overclocked, but i5-4690k can be overclocked. Remember I stated the FX-8350's performance is roughly between a 1st gen and 2nd gen Intel Core series CPU? If you want to get performance that is close to the Core i5-4440 you will probably have to overclock the FX-8350 to 4.5GHz which is possible but not all FX-8350 CPUs can reach it. You need to buy a good CPU cooler and you also need to buy a really good motherboard that allows you to overclock as well. Whereas if you just buy the Core i5-4440 you can use the stock cool and simply purchase an average motherboard. On top of that you do not have to fiddle around with BIOS settings like frequencies and voltages to squeeze out every bit of performance you can. Therefore, switching over to an Intel CPU and motherboard will cost you less money and it will provide you with a little better performance than an overclocked FX-8350 with less hassle.
Of course, if you really feel like splurging you can buy the i5-4690k and overclock for even better performance. But you will also need to buy a CPU cooler and a good motherboard that allows you to overclock. Similar to what you have to do when overclocking the FX-8350.
Whether you want to stick with AMD and upgrade to a FX-83xx CPU or switch to Intel, or not is entirely up to you. You need to make the choice that you think is right for you. AMD is expected to finally release the successor to the FX-83xx / FX-9xxx CPUs called the "Zen" CPU in late 2016 and it will use a new socket; AM4.