You do not need to upgrade anything on that PC except the RAM and a new Windows installation. A new system will be quite a bit peppier, and will boot up faster, but a 3 gig P4 system with a couple of gig of RAM will be just fine for web browsing, and will even run HD movies with a cheap $30-40 video card. I've got a 1.9 gig Pentium M system with 1.5 gig of RAM running HD streaming from Netflix with just a bit of slowness at times, usually it's perfect.
My advice would be to buy 1-2 gig of RAM, and since I'm betting your 3 gig P4 system has a PCIe slot, get a cheap Radeon 3450 to accelerate video. Should be a $100 or so for hardware. Then re-install Windows. You may want to replace the hard-drive, that will keep you from an extra step of backing up and restoring your files, just use the original drive as a secondary and copy your files, favorites back over to the new setup.
$150 with the drive, and you are set. I'm guessing you have not re-installed windows in several years if ever, and are running god knows how many add-ons and spyware in there.