All Macs are a breeze to connect to a network, wired or wireless. Assuming the network at home is working properly, it should be just fine.
There are a couple of things to keep in mind though. If your family's network is "secured" using WEP, (and I say "secured" because WEP is simply not at all secure), then to enter the network password, you'll have to start it with a '$' so the Mac will properly interpret the input. Some network hardware isn't too compliant with standards, and you may need to manually enter a network name (the SSID) to get your new MBA to see the network the first time. Otherwise, you should have a blast!
Oh, and WELCOME to the MacNN forums!