Will this radeon 9250 card
work in my crappy Dell Inspiron 2350? (I only have PCI slots)
What are you wanting to do with the new card that you currently cannot do with the on-board video?
Seriously, a larger upgrade may be more worthwhile than adding more pieces that are limited by the motherboard.
P.S.: Are you certain that it isn't a Dimension 2350
with the Intel i845GL chipset?
Is there a chance that a new socket-478 motherboard based on the i915 chipset with a PCI-express X16 slot could replace your existing motherboard? There are many inexpensive PCI-Express X16 video cards available that perform much better than any PCI video card.