Hi Linda,
RAM is an important factor, particularly when you have multiple applications open. I don't see why it would have caused such frequent blue screens, but I suppose running Windows along with several instances of Internet Explorer and so forth would have made your machine very slow indeed. What operating system have you got? If it's pre-XP, then that is most likely the cause of your errors - As compatible as Windows 98 and ME were, they had plenty of bugs that cause more than the occasional headache. XP is generally a very stable operating system, so if you have that then it makes the blue screens even more difficult to diagnose.
That said, if you are looking for a new PC now is as good a time as any to purchase one as hardware prices are particularly competative at the moment. I generally recommend people purchasing an entire machine to look at somewhere such as Dell, who not only build good quality machines but also back them up with warrenties. You can get more speed for your money, but few have the build quality and service to back it up and if you rely on your PC, that's what you really want to see.
Standalone cards are better than onboard for a number of reasons, though the ability to be flexible is my particular favourite. Again, you might spend a little more but it will allow you to upgrade in the future and prolong the life of your machine - crucial if you are on a budget.
If you want to look at individual items or machines, try
Ebuyer - Cheap Computers Laptops Digital Cameras Televisions or
dabs.com - PC Hardware, Components, Software, Digital Cameras, MP3 Players. Alternatively, look at manufacturers such as Dell to see what you can get for your money. It may be possible to extend the life of your current PC, though in general I'm afraid all but the very top end machines are pretty much out of date within 3 or 4 years. If you can afford it, buy a new low-end machine and look at adding extra RAM and hard drive space. It will keep you going for a fair while yet and will also be much, much faster and (hopefully!) more stable than your current machine. You needn't spend a fortune as there are some good deals to be had.
Best of luck and as usual, if you have any questions, don't be afraid to ask.
John