From what I hear/read, nothing will use all of the cores in a 4 or 6 core processor
Oh? Where did you hear that? Got a link? For starters, Windows will use all the cores. And many games do too.
The advantage to using a more powerful processor is over all better system performance. But - whether you can see that better performance is another matter as over all performance is dependent on my factors besides CPU horsepower - in particular graphics performance and RAM, and to a lesser extent, drive performance as well. And if your usage habits are heavily involved with the Internet, then obviously your network and Internet bandwidth are big influences.
Here's where the fights are going to break out! I'm a home user who doesn't play a lot of games, mainly just use the computer for listening to music, watching movies, surfing the internet, balancing the checkbook... Do I want an Intel or AMD processor and do I want an ATI or nvidia video card, and why?
Doesn't matter. I like Intels but others like AMDs. BOTH
make excellent CPUs that will serve your needs just fine. That said, AMDs, in order to be competitive, can usually do it a little bit cheaper. In reality, you can listen to music, watch movies, surf the Internet, and balance the checkbook with an entry level i3 or AMD equivalent and a couple gig of RAM just fine, including playing many games - and that's with an on-board graphics solution. Install an upgraded graphics solution and at least 3Gb of RAM and you can play most games with no problems. For serious gaming, you will need higher-end CPUs and graphics solutions.
As Neil suggests, you need to establish (and tells us if you want applicable advice) your budget. And don't forget a good PC is built on a good foundation which means a decent case and properly sized PSU from a reputable maker. A good case must be solid, sturdy, and provide plenty of cooling options. The PSU must support your hardware and provide expansion room for upgrades (therefore the PSU must be selected last).
Finally, a new PC MUST have a new Windows license. You cannot carry over Windows from another machine. The ONLY exception is if you have a "retail" license and it is not installed on any other machine. Most copies of Windows are NOT
retail versions, but are OEM or upgrades to OEMs and they can NOT be carried over to new machines.