the router plugs into your computer,then the modem plugs into the router. the router has a bunch of ports on the back.

the router has 2 short antena's on it. it sends the internet signal to the laptop. like a radio. depending on how far you need to or want to send the signal will depend on what router you buy.

the batteries are rechargable. just plug in the laptop. they should last 3 hours or more on 1 charge.

you can find inexpensive laptops on craigslist. i wouldn't go the ebay route unless the shipping was low.

my son got a really good dell on craigslist for college for under $200.