I have a question. What is the method of connection between your modem (or router) and your computer? Is it a 10/100 ethernet adapter, or USB, or something else? I'm just curious if this could be some kind of a bottle-neck situation. I'm sure that the age of the computer can have an effect, but I don't know to what degree.
But anyway, on to the suggestions.
I've built my own computer, so there's a lot I could recommend there. But if your looking for a ready-built computer, I'll share some of my brand preferences.
On the upsides, I'm a big fan of Sony electronics, just not their prices. HP seems to be a good brand right now. Many of family members are using HP laptops and are happy with them. I've heard from several people I know that they like Acer. And for the cheap, eMachines seems to have a real good price. Just be prepared to replace a power supply. I couldn't tell you anything about Gateway.
On the downsides, I would avoid Dell because of their past hardware compatibility issues (if you ever want to upgrade down the road) and general customer service. And I'll say this about Toshiba: I will never buy from them because of a customer service experience I had with them about 7 years ago, but they do seem to make a decent computer.
As far as specs are concerned...
Processor: I highly recommend AMD over Intel for general computing. I do a lot of video editing with a Dual-Core AMD at 2.5 Ghz, and it rocks. I think the Quad-core processors are becoming a lot more popular, but unless you've get serious need for it (or just want the bragging rights), I don't think it's worth the extra money.
RAM: You'll probably want to get atleast 2GB just to keep everything operating smoothly. RAM is fairly cheap and it's almost always worth having plenty of it. I'm still operating on Windows XP, and 2GB is more than enough for my needs. If your going to be running Vista, you might want a little more, maybe 3GB or even 4GB. If you'll be waiting until Windows 7 comes out, then I don't know. Probably going to be similar to the needs of Vista.
Network: I'm not sure on the numbers, but if you've got a high speed internet service, you want to make sure you've got a high speed network adapter in your new machine. I think the old standard of 10/100 is fairly outdated now. I believe most newer machines offer 10/100/1000 speed network adapters. Not sure if this may be one reason you're having issues with your current internet speed, but keep it in mind, and keep an eye open for this specification while shopping.
Other Stuff: I think most everything else boils down to personal needs and price range. Size of the Hard Drive, ability to burn DVD's, built-in Memory Card readers, it's all up to you. You mentioned you write and record music, so maybe look for something with a high quality audio card. You might even find that one of those brands is offering a free upgrade on something like RAM or Processor speed.
Good luck, and Happy Hunting!
Muzak