That $600-$700 budget does not get you a laptop that will run 3D games (like World or Warcraft, Crysis, Call of Duty or Starcraft 2) well.
But they will run internet, movies/music and home office programs very nicely.
Dell Inspiron Laptop Core i3 Processor $529.99 Model: I15R-1161MRB looks "OK"Dell Inspiron 15R Notebook Review
HDMI = yes
802.11n = yes
Bluetooth = yes
USB 3.0 = no. USB 2.0 only
Battery life = 4 hours on idle tasks. 3.5hrs on Wifi websurfing, 2.5hrs on DVD movie viewing.
802.11n is the fastest of the WiFi wireless networking speeds to work with your wireless router.
WiMax is internet service over cellular phone networks. It's not necessary if you have normal internet service from your Internet Service Provider (ISP)
You'd need to purchase a cell phone data plan from your cell phone wireless carrier. That could get expensive in data plan overages.