The resolution is 1920x1200 on a Dell U2410 Ultrasharp monitor. The kids may play some games on it (Minecraft etc) but the PC is not meant for gaming.
Before you even upgrade, what is the problem with the HD5800? Is it broken or the kids have complained that it doesn't perform to their liking? I mean do you want to spend $ for no reason? With that out of the way, before going all the way up to a $300 videocard consider some other options:
#1 GTX950/960/R9 380 are a very solid upgrade and could be enough for your kids. By the sound of it if your kids were OK with a 5800 series card for so long (that series came out in 2009), they probably don't need the fastest $300 card like a GTX970/390.
#2 When considering $300+ card, always look in the price range to see if something is almost as fast for way less. Right now we have such a case = an after-market R9 290 for
$240 with lifetime warranty is a solid choice too. Based on your non-overclocked CPU and based on what you described that the PC won't be used for hardcore gaming, I don't see the justification of spending around $300 for an R9 390 or GTX970. With your stock CPU, you will not get any more FPS from a 390/970 over a 290 on that system, so why spend the extra $?
Also, even with a very fast CPU, after the $240 R9 290, there are seriously diminishing returns for performance per each extra $ spent. Keep in mind that an
after-market 290 ~ reference R9 290 on this chart:
Budget is no issue. I want the best/fastest graphics card that is at the limit of what the CPU can handle. I think this is the last major upgrade for this computer short of a new one.
Right but is the upgrade even needed or you just want to buy a graphics card to treat your kids/self?
With all of that taken into account, based on what you have described I think even a 960 4GB or R9 380 4GB is probably overkill as is for games like Minecraft. You might not even need an upgrade until your kids start complaining and the longer you wait, the cheaper GPUs will get for a given level of performance.
