Your laptop should have everything it needs to connect at a hotel that provides wireless internet access. Usually the hotel with provide you with a key number and when you launch your browser, it will ask for the key. When you enter it, your computer will be set up with a profile for the hotel.
With respect to your house, you will need to buy a wireless router. You will have to connect the router to the DSL modem (make sure you buy a router for DSL and not cable). Then you will configure the router and your computer to create a wireless network. The install instructions included with the router will be enough to get you up and running. There are many how to's on this site to help you set up the proper security.
Linksys and DLink are some companies that make inexpensive routers for home use.
I recently purchased a new laptop.
It has "Wireless LAN: 54g 802.11b/g wireless LAN"
Do I need anything else to be able to connect to the internet when I'm staying in a hotel that provides wireless Internet?
My telephone company only has DSL, do I need anything else to connect to the internet at home?