Upgrade hardware individually
If you can determine what part is the bottleneck in your system, a simple upgrade will be much cheaper than getting a whole new system. With laptops, your options here are pretty much limited to RAM, hard drive and wireless card, but desktop users will have a little more room to play with.
If you've got less than 4GB of RAM, investigate adding more — with more moving online, and browsers being particularly memory hungry, having more memory is never a bad thing.
Just be aware that if you have 4GB RAM or more, you'll need a 64-bit operating system in order to recognise and use it all — which for some people still running XP might mean an operating system upgrade as well.
The hard disk is typically the slowest part of the computer. When loading data or if you run out of RAM, your hard drive gets hit — and if it's old and slow, you're going to feel it.
While solid-state drives (SSDs) have definitely addressed the speed issue, they're expensive compared to mechanical hard drives and are much lower in capacity. If you can only afford a mechanical hard drive, make sure to get one that's at least 7200rpm. As a basic rule of thumb, larger capacity drives at the same revolutions per minute are faster than smaller capacity drives.
If you're finding games are a bit slow on your desktop, it may be time to update your graphics card. These days, a new graphics card will have significantly more effect on your gaming experience than an updated CPU will.
External, upgradeable graphics on the laptop never really took off — but perhaps we'll see something emerge should Intel's Thunderbolt interface take off.
If you're finding the wireless in your laptop slow and unreliable, it's actually quite easy to swap in a new wireless card in your laptop — it uses a standard interface called Mini PCI Express. Just be aware of a gotcha: some vendors like HP will only allow certain wireless cards to be used in their laptops for warranty and support issues. Others, like Dell, will accept pretty much anything. If you've got a laptop that doesn't like alternative wireless cards, you can always get a high-end USB wireless adapter that should improve performance greatly.
If you've got a friendly laptop, it's a simple case (which, depending on the laptop, may involve removing layers of equipment, admittedly) of disconnecting the antenna wire, removing the old card, putting in the new one, reattaching the wire and then installing appropriate drivers. The hard bit is finding someone that will sell you a decent chip.
Also consider the age of your router — wireless technology is improving all the time, and the more expensive routers generally do provide superior wireless performance. Just remember there's no such thing as a good router brand — stick to models that have had a good response in reviews and online.