Laptop newbie, non techie - HELP!

My normal home PC died the other day, and after taking advice and trying all sorts of restore processes, it appeared to be beyond recovery. I decided to get a laptop instead of a new fixed PC, and was encouraged by the sales person in PC World to get a Wireless Router so that I can use it anywhere in the house. She took one look at my old modem, which I'd taken in to ask whether it was faulty, and pronounced it obsolete!

Anyway, I wanted to transfer my Wanadoo broadband contract to the new machine, but the install CD always requires "connect the modem" at a certain point. I even phoned Orange/Wanadoo for advice but was told that I have to use the original modem, so I did that and got the Internet up and running, hence I'm here now!

Stupid question - having installed the Wireless ADSL Router, should the Internet still connect if I disconnect the modem lead from the USB port on my laptop, 'cos it doesn't?

Next stupid question - have I been misled about whether a laptop not linked to a static computer can actually connect to the Internet via a wireless router?

Worrying question - on logging on, I'm getting a message to say that I am in range of three Wireless Network Connections, and the signal is excellent! This is worrying because I am currently still connected to the Internet via my old plugged-in modem, so my own router is not connected to my phone line. Also, one of the three connections on offer is apparently my city's football club, which is about 2 miles away! The other two have numeric references, so goodness knows where they are.

I need to go into my online bank account very soon, but am nervous in case I'm linked up via someone else's system :eek: .

This post probably bears out the theory that stupid people should not be allowed near computers, but I would be most grateful for any replies and advice, in as basic terminology as possible!