If you just need to play flash games and surf the web, cheapest way to go is to build a DIY pc. Tigerdirect and Newegg are good places to start looking. Newegg even has a tutorial video on how to put them together. It's pretty easy, saves money, and you can use your old OS if you have the disks or buy a new one.
This is the way I always go, i.e. building my own. This is only for those that are confident enough to do so and if one has ever built one before it no longer seems complex like it does for those that have not build one. You will not get an OS this way unless you purchase a DIY kit though. If you buy the parts and not a kit you need to be aware of the power requirements of your components (online tools to calculate what power supply wattage you would need) as well as which cpu matches which motherboard slot, what memory is qualified for what motherboard, etc. It isn't quite as rough as it may sound but it does involve some research (which I find most enjoyable -- but I suppose not everyone would.
You can go to sites like tomshardware.com (as well as others) to find current reviews of products and forums where one can ask for recommendations on custom builds or opinions on components. They also have articles they update frequently showing the best for the money across a variety of components and prices (e.g. best video card for the money from the lower range to the upper, best gaming cpu for the money across such ranges, etc). They even point out facts such as when an 'upgrade' in tier may cost $200 more and yet only yield a 5% improvement.
If you purchase a branded computer you will tend to get a fair deal but you will suffer from extra unneeded bloatware (software that you have no use for and should uninstall), lower quality components (in most cases but their can be exceptions), an inferior power supply (in almost every case), limited expandability and connectivity (but often isn't needed for such consumers), poor video card or reliance upon motherboards onboard video or the CPU on chip video (an additional gaming card can generally be purchased and easily installed -- but make sure your power supply can handle the demands of this card or you may have to swap that as well), a poor computer case that has no dust filtration and poor circulation and cooling.