Well, firstly, you do not NEED a wireless router, as you could simply connect the modem to the laptop via a cable (typically USB), install any drivers for it and off you go. However, as wireless routers are pretty cheap these days, you would normally use one of these instead. Also, most of them include an ADSL modem, so you would discard the existing modem, physically connect the main PC to the router via a LAN cable (normally included with the router, else a couple of quid), and connect wirelessly to the router via the laptop as Ethel suggests.
Also, regarding the laptop issue when "away from my house", as Ethel says, there are hotspots for which you normally need to pay as you go, and then there are standard wireless networks, like your friends and neighbours may possess. If you go around to a friends house and want to connect to their wireless network, if it is unsecured (hopefully not), then it is dead easy, simply click on the wireless icon in the system tray, then View Wireless Networks, choose the listed network and connect. Dead easy. If the network is secured, just a bit more work involved as you will need a key (a password effectively) which your friend can give you. Also, if they have turned SSID broadcast off, you will not see it in the list of available wireless networks (it is INVISIBLE), however, fear not, it is there... you simply have to add it manually (click the wireless icon again, then Properties, then the Wireless Networks Tab, then Add and off you go...
Good luck.