One thing at a time! When you connect to the internet at home is it by a dial up modem or broadband adsl (BT phone line) or Broadband Cable (NTL/Telewest)? If your laptop has built in wireless you could connect to the internet via a wireless router modem. Your other PC would also have to use a wireless USB adapter OR have an internal card fitted for an ethernet (cable) connection. It may have one already - do you know?
On the subject of travelling with your laptop let us know which make and model it is. If it has an internal modem, most have, you can simply use your current ISP`s dial up facility on a pay as you go basis within the UK. (You pay for the phone line time wherever you are) Hotels charge a lot, friends probably won`t. When abroad you can increasingly connect wirelessly to "hot spots" at airports and Internet Cafe`s using your laptop or simply buy time on one of their machines to surf the net and look at your e-mails by webmail. I do it all the time.
Post back with some answers or ask if you don`t understand anything I`ve said. I`ll put you some links to illustrate what I`m on about.