Hi and welcome to LQ
Do you know how you obtain an IP? If not, just make a call to your ISP and ask em, I'm sure they won't have a problem telling you, regardless if they support Linux or not. For direct answers to your questions though...
Local IP adress of machine= ?
What should I be typing in here for my wireless network?
Also what adresses should I type in at DHCP and DNS Server?
How will I know if these addresses even work? Will something pop up to tell me?
Local IP = 127.0.0.1
Remote IP = (Depends on if you get a static IP from your ISP or a dynamic..)
If you are obtaining everything via DHCP, then you shouldn't be having to type anything in any of these blanks, it should all be configured via dhcp for you, that would include the DNS settings. Otherwise, you'll also want to ask your ISP for the IP's of your DNS servers you are to use.
You'll know they work because you'll be surfing all over LQ from your couch
You'll also know they work, because you'll get them from your ISP, if indeed you are required to supply them (again, if you obtain your IP info via DHCP, which your ISP should tell you, then you shouldn't need to worry about any of these settings...).