You can definitely network with a PC notebook (or any other PC computer, for that matter) via wireless card without a router. Wireless networking through WiFi is an international standard supported by not only all manufacturers, but by every modern operating system. No sweat.
Connecting wirelessly without a router is called establishing an "ad hoc" or "computer-to-computer" network, and set up is fairly simple. The problem is that it won't be the kind of network you can reconnect to when you want to; you'll have to reestablish it from scratch. Remember that "ad hoc" means basically "as needed." It's not intended to be a consistent, permanent network. It will allow you to share files and printers, and with a little manuvering share an Internet connection, but it's not supposed to be a permanent solution.
The good news is that a wireless router can be inexpensive, they are not at all hard to set up (with ANY computer that has a browser-no matter what you might read anywhere), and once set up properly they can be almost completely ignored.
Here's a very reasonably priced wireless router, the Netgear WGR614, which comes out to less than $30 after rebate, and has free shipping from Buy.com. Considering the benefits of sharing an Internet connection via wireless, this could be just the deal to get you started.
As for Bluetooth, there are pluses and minuses to that technology, and while it's probably possible to effectively network using Bluetooth, since it has absolutely ZERO security (it's meant to connect with peripherals, so who needs security between your computer and your printer?) it is a very poor choice.
