Your new laptop should automatically talk to the wireless router (via its own wireless card) and also connect to any other wireless networks you have access to; either free ones like in train stations/hotels or ones where your friends give you the password. For connecting on the road where you have no access to a wireless network, then you need a dongle.
Most laptops nowadays will have lots of memory and be capable of everyday use. You'll only need a certain spec if you're specifically into gaming or movie watching.
I'm sure you'll get some recommendations for specific brands/specs, you should have loads of choice as the shops are desperate to sell at the mo!