In Firefox, Options -> Advanced -> General tab, do you have "use hardware acceleration when available" checked or unchecked? This could affect rendering of fonts depending on the capability of your onboard HD4250, and you might try the alternative to see if that helps or further hurts.
Of course if you have a similar problem in IE it might be a similar issue there.
What version of Catalyst driver do you have installed? As was mentioned already, having current drivers installed is almost always the right thing.
What version of Firefox and IE do you have installed?
There used to be a problem caused by a Microsoft update to IE (KB2670838 I think) that affected font rendering in Firefox when using ATI video. The workaround was to disable GFX.CONTENT.AZURE.ENABLED -> FALSE in ABOUT:CONFIG with Firefox. But I think this has been corrected, although my current Firefox setup shows this still as FALSE (and I'm not having any font rendering problem). I believe this setting is involved with "hardware acceleration" from the video card. Anyway, you might try toggling your setting to see if it helps or hurts.
Finally, you might think about installing a somewhat stronger video card. I see that your motherboard has an PCIe x16 expansion slot, so any decent low-power video card would definitely be an improvement over your onboard HD4250. For example you might consider an HD5450, HD6450, HD7750, etc., either in low-profile (if your case is small form-factor and requires low-profile cards) or full-size.
These three cards are modestly priced (and with increasingly better performance) and do not require external power, and come in low-profile with two brackets, so you can swap a full-size bracket onto a low-profile card for use in a standard mid-tower case which takes full-size expansion cards. Just a thought, as all three of these cards run with the very latest Catalyst drivers, rather than the "legacy" driver that your HD4250 uses.
