It should be possible to configure your router to permanently connect to the internet, or connect automatically on demand. I would recommend using these settings - after all, that's why broadband is called "always on". The router should also cache your password details.
If your router is the USB kind, then it's a bit more tricky, as these routers are actually not really routers at all, but ADSL/cable modems. They are just designed to connect a single computer to the internet. However, I suspect that is not the case as you talk about connecting with your browser.
Also, the fact that your wifi isn't connecting properly rings alarm bells. You need to make sure you've got the security set up properly, that you're using the correct passkey, and make sure that you don't need to press a button on the router when you want to allow other devices to connect.
Really a lot of this stuff is dependant on the make and model of router, and I think this I really more a question for your router manufacturer.