Hi Eric,
The simple answer to your question is a new machine will be seriously faster than your old one. I guessed you don't know enough to upgrade things yourself. The main trouble with old machines is you first have to identify the bits inside then you have to track down a shop that sells it.
Speccy is good software and will tell you, if you want to try upgrading, but that wasn't your question.
You have several options:
a) buy new from Curry World
b) upgrade the bits of your machine
c) buy online
The answer/choice depends on how much money you have, and what you want your computer to do.
If you only want it to surf then it may well be better to get a tablet of some kind. The Google Nexus is just under £200.
If you want to do more computing tasks (simple letters, surfing, the odd spreadsheet) a netbook will do fine is about £300 tops. Don't spend more on a netbook.
The final option is a laptop. I'm using a Lenovo and it's brilliant and I push it insanely hard for work. It was just over £400.
Then you have desktops. They will start at £400 for a decent one but have plenty of power. I'm sure anything will feel like a Porsche compared to your old one.
Having said all that, nothing, not even a dual Xeon super computer will help your broadband. It will launch IE instantly, but then the waiting will start if the connection is bad. I only have a 2MB connection despite it being sold as up to 8MB and light internet is fine, and watching BBC iplayer works most of the time.
As for upgrading, I would only touch two things - memory and the disk, in that order. You have a sneaky cheap option. If you install a brand new Windows if might breath life into the old dog. Windows 8 comes out next week (26th Oct) and is £49. It is very different from XP though, so you might not like it. Also you need to check your machine can even install it.
The same applies for Windows 7 which is about £80 if you buy the OEM version.
Finally, there's the RAM and hard disk. RAM is dirt cheap but your machine's slots may already be full. If not, great, but you need the right stuff. It's in lots of different styles but I'm guessing you will have DDR1 or DDR2 (think Cortina MKI and MKII). Speccy will tell you - just get more of the same. Sadly the older RAM is the more rare and pricier it gets. It's about double the price of DDR3 right now - £20 for 2GB.
For the hard disk, it's easy enough on paper. It will be an IDE disk you need. Check Speccy again just in case. You will also need to install Windows on it and use the old one as a spare. They are getting very old too though - ebuyer only have 5 on sale.
As for fitting any of this if you go the upgrade route, you just need a screwdriver and plenty of light. It's like lego inside - everything is plugs and clips and easy enough. RAM will only fit one way round so it's not possible to mess up. The only mistake is you don't push it into the slot firmly enough and not happens when you try to boot.
Bottom line: Upgrading is cheaper but a hassle.
Personally, I'd choose a Windows tablet when they come out if I had £450 spare or the Nexus if I had only £200.
I hope all that makes sense and helps you decide.
Mike