The problem is jessterman, I only have a 300W power supply. I don't want to invest too much money into this computer so I don't want to buy a new PSU. That's why I was thinking of the R7 240.
Actually, the problem is that 300w is much more than you need for your PC including an HD 7750. That card will not draw more than 55w when overclocked and running at full-load, so adding the rest of your power draw, you will never use more than 150w with that setup. And any PSU will be able to supply that amount of power - even tiny 220w ones - check the reviews on Newegg. Also, check Tom's forums for the hundreds of threads with "300w PSU video card" in the title. They all have the same answer.
The "minimum PSU wattage" listed in the HD 7750's spec sheet is just to cover AMD's butt - so if someone decided to put the card into a 10-year-old workstation with an overclocked 140w CPU and 10 Velociraptor hard drives, and it pushes their feeble PSU past its breaking point, they can hide behind their disclaimer of "at least 450w."
Bottom line is, you will be just fine putting that video card in, and you will get at least double the performance of an R7 240 for only $20 more. Buy it, love it, welcome to PC gaming.