Intel has recently become very confusing on their naming.
i7 is socket 1156 and socket 1366, quad core, uses triple channel DDR 3 memory, Hyperthreading
i5 is strictly 1156. both dual and quad core, dual channel DDR 3 memory, no Hyperthreading
i3 is strictly 1156, only dual core, no hyperthreading
best way to look at is the numbers assigned to the processors:
900 series will be i7, socket 1366; all are quadcore; hyperthreading; i don't think they have turbo
800 series will be i7, socket 1156; all are quadcore; hyperthreading and turbo
700 series will be i5, socket 1156; all are quadcore; hyperthreading, no turbo
600 series will be i5, socket 1156; all are dual core; hyperthreading and turbo
500 series will be i3, socket 1156; all are dual core; hyperthreading, no turbo
I'm not sure how many of the 600s and 500s are out yet.
Advantage of the 700s and 800s is that the socket 1156 motherboards are noticeably cheaper.
The i5 and i7 use DDR 3 memory. Not sure about i3.
Best bang for the buck: i5 series by most estimates.
If you are going to change boards, I would automatically ditch the Dell case and power supply.
Quad cores don't necessarily equate to a faster PC, depending on what you want to do with it. Many applications can't properly take advantage of all 4 cores and a dual core may be faster at any given price point, depending on your intended usage.