Dell Inspiron 580 - Core i3 550 3.2 GHz - 4 GB DDR3 SDRAM - 1 TB, Hyperthreaded, integrated graphic.
CPU is not terrible, its a dual core with hyperthreading so looks like a quad core. 4GB Memroy is adequate and if your windows is 32 bit then its all you can use.
Best upgrade is adding a video card. Huge improvement over integrated video, low cost, works with your existing MB, CPU, Memory, Power supply. Easy to install.
The looks like a mid tower (6.9 in x Depth 17.4 in x Height 14.8 in) that takes a full height (not low profile) video card.
I can't figure out how much power you have. A GT 740 is apt to work, at 64W TDP (you have a x16 pcie slot, that should reserve 75W from your PSU). A GT 730 $60US is a good bit slower, but uses only 38W. Both will absolutely kill your current video for gaming.
If you can open case and tell us the wattage of power supply that would help. It's usually on a yellow sticker you can read with the case open. Or mabe you can find the PSU output by googling. I think either video card is low enough power to work with your current power supply, but would like to know the PSU output to be sure.
Here is a compare of GT 730 vs GT740
http://gpu.userbenchmark.com/Compare/Nvidia-GeForce-GT-730-vs-Nvidia-GeForce-GT-740/m12582vsm13294
Here is a compare of your current integrated graphics to the GT 730. 730 will be 4X faster.
http://gpu.userbenchmark.com/Compare/Nvidia-GeForce-GT-730-vs-Intel-HD-Graphics-Clarkdale-073309-GHz/m12582vsm7702
If you go to youtube you can find videos of people playing Skyrim, SFM, Counter Strike etc with GT740 so you can see framerates and video quality settings.
(The only other upgrade is to replace CPU/MB/Memory which will be a couple hundred and not worth it vs. building a new PC from scratch or buying a later refurb and updating it, or even getting a $350 AMD A8 based laptop).