In the last month or so I've fixed about 4 separate cases of computers not being able to access the internet. I thought it was worth a brief notes on the different cases, and the solutions, to help others who hit problems.The various problems:

Neighbour's laptop - apparently couldn't do anything.
Dad's PC - couldn't browse the web, but one (and only one) email account was working.
Martha's Vista laptop - apparently could connect to wifi network, but couldn't do anything on the internet
my iPod Touch - could connect to wifi network, but couldn't do anything on the internet

My general approach was the same in all cases:

What does the system think it's connected to? For this I use ipconfig /all (Win) or ifconfig (Mac) to look at what network adapters are active, the IP addresses, the DNS servers it's trying to use, and the gateway addresses. Some experience is needed here to work out what these imply. I'll try and post a flowchart here at some point that helps with this.
If it's connected to a broadband router OK, is everything not working or just some services? Here I use something like ping www.bbc.co.uk to see if basic DNS lookups are running, and ping 212.58.251.195 to see if basic TCP/IP services are connecting out through the server to the internet.
If these are OK, which higher level services are working (if any)? In Dad's case, just email, but in all the others', nothing at all.
Is there some sort of filter getting in the way? These could be "proxies" designed to help connections in/out organisations with restrictive networks (eg, schools, colleges) or firewalls being too aggressive or playing up.

In the neighbour's case it was a proxy setting that had got turned on (which she then remembered about), which was redirecting internet requests to something that didn't exist. A quick look under Control Panel > Internet Options > Connections found it and turning it off cured it immediately.For dad it was his firewall misbehaving; turning that off restored the internet. A quick reinstall has hopefully cured the problem.

Nowhere near as easy for the laptop and iPod problems at home. One Mac laptop was fine on the home wifi, but the Vista one wasn't, which made it sound like the laptop. A full hunt around showed that it was connecting OK to the router on wired or wireless, but all internet operations were failing, despite no proxy or firewall problems. I was stumped. After much fiddling with some advanced ideas on dynamic allocation of receive windows and netsh tcp global configuration, it still wasn't working. Grrr.Reluctantly I looked again at the Sky router. All settings seemed fine. I tried turning authentication on and off, and fiddling with different wireless settings - all to no avail. As a last resort I tried nuking the box - restoring the factory settings. Bingo! the laptop and iPod then worked! This shouldn't have worked, but clearly shows that the Sky Router can get itself into a state where it doesn't service all connected clients.

Resolving internet connection problems