The first charge is approx. 3 hours - it tells you at the start of the manual.
You get a green flashing light as a notification to tell you you have a text, or a Twitter update etc. or if the phone wants your attention. If you've put it on charge the light should be steady orange while charging, then steady green when it is charged, but still leave it for at least 3 hours. If the light is flashing it sounds to me as though you've switched it on and are charging it while switched on.
If the phone isn't charging when plugged into the mains but charges via USB it sounds like a faulty plug. Take it back to the shop and ask them to change it.
Regarding the internet your SIM card configures the phone's internet settings regarding 3G etc. The phone's setup routine asks if you want to connect to a wireless network. If you have one say yes, and that way you're configured for at home and on your travels.
A word of advice: Be prepared to charge your Desire a lot. I have one and it is brilliant, but I have to charge it every night, and I generally have wi-fi, mobile internet (3G), GPS and Bluetooth all turned off all the time, apart from for a few hours in the evening when I'll turn wi-fi on. Add widgets to your homescreen to allow you to turn such things on and off with a single press. Ultimately the more you use the phone the faster the battery runs down. I don't use the radio, the MP3 player, and most of the time it is in my pocket for emergency calls and maybe ten or so texts a day. If I leave it alone and don't use wi-fi at all the battery can last me two or three days - ultimately anything network-related (wi-fi, mobile internet, Bluetooth) saps the battery. Tip: Buy a spare charger - a cheap "universal" one from the supermarket will be fine - and keep it in your bag / desk so you can charge the phone when you are away from home if need be.