Yes you can player multiplayer online with friends..games like Mario Strikers Charged and Battalion Wars 2 have online with friends, but require you adding his game code, and then him adding your game code. Don't confuse the codes with your Wii number, the game codes are found in the game.
Other games like Medal Of Honor Heroes 2 and NBA 08 don't require friend codes, but this way you can't play with friends..rather random people.
To get online:
Well do you have a wireless internet connection?
If not you can do one of two things.
If your Wii and computer are close (within 5 feet) you can buy a LAN adapter which can be bought from a Best Buy or most retailers. What it does is connects your Wii to your PC, it comes with instructions and is easy to install.
Or if your Wii is not close you can buy a Wi-Fi usb connector which can also be found at most video game retailers. This acts as a wireless connection and it plugs into your usb drive on your computer, sending a signal to your Wii allowing for Online play of whatever kind. This is also easy to setup but may require you to turn off your firewall when connecting to the internet on your Wii as it sometimes blocks your signal.
And if you already have a wireless internet connection you simply just need to search for an access point in your Wii internet settings.