A lot depends on what you mean by [quote]home purposes[/quote]:-}
If you mean internet, office, music, 9in fact anything except intensive gaming and video processing) a laptop will be ideal. A laptop for gaming and video processing is also possible, but not cheap.
You can use your broadband connection around the house by getting a wireless router with built-in modem. I don't know the details of what will be needed, but someone else will.
A good option is a decent laptop and, say, a 19inch TFT and external keyboard. Together with an external hard disk for back up purposes, you will have all that you need.