HELP! Need to connect wirelessly

I thought I understood this by reading many many threads on this subject but I am sooooo confused.  I have a Dell desktop that has a built-in modem and it connects to the internet by dial-up.  (Yes, I live in the hills. I have no options when it comes to this) Now, I have a Dell Vostro 1710 that doesn't have a built-in modem.(Surprised me).  I want to use it around the house anywhere, wirelessly.  Do I have to buy a modem and a router?  I see some routers have built in modems but will they work with dial-up?  The routers really confuse me.  There's g, n etc... What do I need?  How do I connect  it?  Desktop-----router-----modem------laptop? How?  Thanks