That laptop has wireless networking built in, you don't need to add a separate Wireless Network Adapter.
You don't *NEED* a router to share the broadband connection, but if you do buy one be sure to get a "stand-alone" router, often described as a Cable/DSL router. (Don't confuse the reference to DSL with a combined "Router/ADSL Modem" which is *NOT* suitable for use with Cable broadband.)
To create a wireless network for Internet access and File/Printer sharing, you have essentially two options:
1) Install a Wireless Network Adapter in each PC and create an “Ad Hoc” Wireless Network. This is a network in which the two PCs talk directly with one another, rather than via a Wireless Access Point. “Internet Connection Sharing” is then enabled on the actual Internet connection in the “host” PC and the “client” PC will have Internet access provided the “host” is running and online.
When you enable “Internet Connection Sharing” on the Internet connection, it automatically configures the Local Area Connection with the IP address 192.168.0.1 and subnet mask 255.255.255.0. However, as you will be using a WiFi adapter, you want this address to be allocated to it, rather than a “wired” Ethernet adapter if you have one fitted, so the Local Area Connection should be disabled before setting up “Internet Connection Sharing”. The "Client" PC's Local Area Connection should be set to get its IP address automatically, which it does by DHCP from the "Host" PC. Also, if you run the Zone Alarm firewall in your “Host” PC, the Internet Zone Security level must be reduced from High to Medium or “ICS” traffic will be blocked.
To enable “Internet Connection Sharing” in Windows XP, open the Network Connections folder, right click the actual internet connection (modem connection), select Properties and then the Advanced tab. Tick the box “Allow other network users...etc”.
2) The preferred option for connecting more than one computer is to use a broadband Router and, with ADSL, this should be a combined “Router/ADSL Modem”. This is an ”Infrastructure” wireless network because the connected PCs communicate via the router’s Wireless Access Point. Again, a wireless adapter is installed in any PC that needs to connect wirelessly, but often the main PC is close enough to the router for it to use a “wired” connection with only the remote PC connecting wirelessly.
With a router, it is the router which connects directly to the ISP and not one of the connected PCs. The router in turn allocates the IP addresses to the PCs, using one of the address ranges reserved for Local Area Networks, usually 192.168.xxx.xxx
Either of these options will additionally allow you to run File and/or Printer Sharing wirelessly across the network.
If you are running software firewalls, the IP addresses of the networked PCs must be put into its "trusted" area to allow them to communicate with one another.