but you are going to be disappointed with the rest of the answer.
Yes, you can take that machine of OS X but not Tiger. Best choice of OS X for that machine is 10.3.9 (Panther). While it is possible to go to 10.4 (Tiger), you will find that the machine with only 256MB of RAM and a 333Mhz processor will leave you waiting around for something to happen.
Yes, you can use wireless internet on that machine but, you will need to find a USB wireless adapter or search the internet for an iMac Wireless Adapter and an original Apple 802.11b Airport Card.
http://www.ebay.com/itm/Original-Airport-Card-w-adapter-for-Apple-G3-G4-USED-/140909311742?pt=US_Laptop_Network_Cards&hash=item20ced976fe is what you are looking for. NOT an Airport Extreme.
The problem with finding a USB wireless adapter is that most of the available ones run on a higher version of OS X than your machine does.
It could be a fun project but remember that 10.3/10.4 are not going to be mainstream. The Web browsers available for those versions of OS X are hopelessly out of touch with a vast number of websites.
i just picked up a G3 Grape Tray Loading iMac for free and i was wondering how i could update it to OS X tiger and also use wireless internet on it