If your Verizon DSL plan is for one computer only, check to see if they allow multiple computers to be used, otherwise it might be a bit more difficult to set up a router because they wouldn't want you to
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To answer your other questions, a standard NIC does not require any drivers under XP. Using a DSL modem with the ethernet port does not require any drivers either. However it is neccessary to login to verizon to be on line using your user name and password. Don't worry, this only has to be done once. This is called PPPoE. It used to be that the PPPoE software would run on your computer, but now usually, you log into the DSL modem to put in your user name and password and it handles the PPPoE. That's how my Verizon Westell 2200 works. Verizon does send a CD-ROM with software to set up the account, it installs some help software, but no drivers or anything, the whole process is very smooth.
If you will only be using DSL with one PC, a router is not necessary. If you want to use multiple computers or to have wireless, a wireless router or the Westell wireless router/modem combination will be needed. If you want to go that route, I would suggest the Westell combination unit because all of the instructions will be provided on the CD, and if there are problems, they will be able to handle them easily. Also, it reduces the amout of cords and plugs you have.
If you choose not to get the combination unit, you will first set up the DSL to one computer, and then you will add the router later. Settings changes may need to made to the modem, and then the router is added and set up. There are guided instructions that show how to do this from the Verizon website. You can find them by searching the help page
Here for the model no. of your modem (ie 2200). I would reccomend a Linksys router because that is what Verizon shows in it's instructions, because they used to give those out. It may not be the exact model you have, but it should be the same process.
Hope this helps.