I've read the 7800GS has a 6 pin PCI-E power socket, but comes with an adaptor in which you plug in two molex connectors. In the above link is the connectors my power supply comes with, would I use the IDE 4 pin connectors to power it?
Your power supply wont come close to running either of those cards. the fastest AGP card available is the ATI 4650. This card does not use a power cable at all so should be fine. there is always the possibility you may need to upgrade the Bios for comparability, its something worth checking which ever card you choose.
You may or may not need to update the bios , if you go to your motherboards web site where you would get the bios update and see where the current bios you have falls then you cans see how many updates there have been since. You can also probably see what the bios updates consist of and what if any compatibility issues they may have addressed.
Reading the review on this website, ATI 4650 needs no additional power if it's a PCI-E. I'd however be buying an AGP which does require additional power. To show I'm trying to do my homework, a quick Google search reveals illiterate offerings stating 300w isn't enough: http://answers.yahoo.com/question/index?qid=20090422131...
Gah! I hate this technical malarkey. I know more about graphics card which my computer can't handle than what it can!
Emachines uses pretty crappy components, that 300 watt PSU probably puts out 250 at most at load. Probably lower.
I would suggest looking at a more modern used computer, a 3-3.2 gig P4 with a PCIe slot that can run a current video card should be around $100. Then for about $40-50 you can pick up a video card to run in it. Or maybe even find one that already has it.
I suggest you look for older workstation towers, like IBM IntelliStations, they have 350 or so watt power supplies and good components. I've had good success re-building them as low end gaming computers.