As a computer technician/guru, i strongly recommend to upgrade/rebuild a new pc. Sure it might sound nerve wracking at first, but you will benefit on newer parts than the old ones.
AMD Sempron is just for basic web surfing, checking emails, chatting with friends, watching videos online that are not cpu intensive. Sure it can probably run modern games, but its not going to give you the best gameplay you want. I think Sempron is similar to a Intel Celeron, just has a bit more power.
As for building a new pc, you dont need to buy all parts new. First get a motherboard that supports pretty much all the latest components, PCI-E, memory slot supporting DDR2, some IDE ports, couple of sata ports, supports dual and quad cores, etc..
Then get a cpu that can handle intensive games and videos. There is a lot of cpus out there, so the only way to tell what cpu is good and in your budget is to go to
http://www.tomshardware.com/charts/processors,6.htmland it will show the benchmarks for each cpu. You can choose what type of benchmark, gaming, video encoding, etc.
Now comes Video card, here
http://www.tomshardware.com/charts/graphics-cards,1.htmlyou will need to do extensive research to figure out whats the best video card is out that fits your budget.
Here is the hard drives
http://www.tomshardware.com/charts/2009-3.5-desktop-hard-drive-charts/benchmarks,50.htmlif you have a hard drive that spins at 7200 rpm and has enough space, you prob wont need to buy a new one.. But a faster HD makes a difference in the end.
Lastly is the power supply. You dont want to leave out this critical component. Get a good brand, quality, and price power supply. Do NOT get cheap ones, as the fuse/capacitors inside will burn causing a shortage that might short your entire components.
I tested a power supply at my friends house, and his was a generic crap that died out. I gave him a list of brand names that are good and quality wise are decent.
I tell to all of my customers, do not buy cheap components. I love them buying a cheap mobo, and then coming back to me calling me, the mobo doesnt work..
As I said earlier about not buying all parts new, dont buy a used powersupply or a used mobo, those two should be new and at a decent price, include the memory too.
Check out craigslist.org and look for video cards that people might sell cheap. Hey I mean you can try to find new parts that people buy and never returned, so they sell them, but make sure its never been opened, tampered, used...
I buy used computer shit before on craigslist, like this lcd monitor im staring at.. its a 19 inch shit brand for $40.. This lcd goes for $100 over.. works great but makes popping sounds like pop corn. The capictors are blowing up one by one inside, and It still works fine loll.
Good luck.
My fav mobo brand is Gigabyte..