The computer is a decent deal for $400. It will run older games at decent settings, but will definitely have issues with newer games.
If you want something more, this is what has to be addressed:
1) What resolution are you playing at. This is a reference to amount of pixels that need to be outputted. Older monitors often have smaller resolutions, and thus don't require as much performance to drive. Common resolutions are 1920x1080, 1024x768, 1440x900, and others listed here:
http://en.wikipedia.org/wiki/Display_resolution
2) What power supply do you have. There are two things that need to be addressed here. Do you have enough wattage, and do you have the appropriate connectors.
Wattage is tricky. No name, or even large producers, often skimp on quality power supplies. They could be "rated" at 430 watts, and only produce 360 stably. A decent new power supply will likely run you between $60 and $140, depending upon features and rated wattage. You should always be willing to spend extra on the PSU (power supply unit), as it can fry an entire system if you cheap out on it.
Connectors are fun. Modern PSUs will always have 24pin, 4 pin, molex, and SATA connections (Motherboard, motherboard, fan/old drive, and HDD/SSD/optical drive respectively). What they may not have is pci-e pins.
Modern middle to high end graphics cards need extra power (above the pci-e bus they are connected to) in order to run. Both 6 and 8 pin connectors are provided in a PSU to deliver this power. You need to have an idea of your graphics card's needs when deciding which connectors you'll need. Given your lack of background, I'd recommend a 6 and 8 pin connector as minimum requirements. You'll need four 8 pin connectors to run two high end
graphics cards, assuming you've got a thousand dollars to spend. This is always difficult to give recommendations on, as future expansion is nice, but immediate costs get high with more connectors.
3) What kind of graphics do you want? At low resolutions (part 1), you can have higher settings than on higher resolutions, assuming the same card is used in both scenarios. Let's start by looking at what you have:
The 5570 is a productivity card. By this, I mean it was not for gaming, but for basic graphics for office work. It won't be able to hold its own versus even mild gaming.
Looking towards playable gaming, you should start with the 67xx series of AMD cards. You might look at Nvidea (550 Ti series), but at your price point AMD is a better fit. Find something like an HD 6770, which should not need extra power connectors (anything better will need them). You'll get playable frame rates from bf3 with lower resolution and maybe medium settings with that card. The near $100 price tag shouldn't hurt too much either.
If you upgrade the PSU, and want a really pretty experience you can shoot for either an AMD 6850 or Nvidea 560 series card. Both come in around $150, and offer substantially better gaming performance (assuming you have the pci-e power connectors from the PSU). Of course, you'll be looking at putting down another ~$250 into this computer to make it gaming ready with both upgrades.
4) A new rig would cost about:
OS: 100
Motherboard: 100
CPU: 200
RAM: 50
Case: 50
GPU: 150
PSU: 60
HDD/ODD: 70
Total: 780 for a very good gaming rig
If we were to rebuild something capable of what you'd want we'd wind up at about $600, and wind up with better performance. Unless you can get money for selling your old PSU and GPU (graphics card) you're looking at an expensive proposition of getting upgraded from an older rig. None of this even touches the older motherboard and other potential flaws (ddr3 ram means next to nothing, 1 TB HDD mean nothing, so we can't conjecture as to performance of these parts).
What I'm trying to say, while being as little of an ass as possible, is that there are far more questions here than we can adequately answer. Some of what I have said is based upon conjecture, and other bits are based upon potentially faulty assumptions. In the future it may be more useful to provide a link to what you've bought (all HP products with model numbers have product pages on HPs website), and ask questions after that rather than listing parts without adequate information, then asking a rather blanket statement of "is this good?" We can offer more help if we know where you're starting from.