If you don't have a Wi-Fi connection at home/school/work, then you will either need to find a friend who does have it, or see if there are any public Wi-Fi hotspots in your area. If you don't live in a city, then it is unlikely you find public Wi-Fi.
The alternative is to buy a Wi-Fi router. You can connect your computer into it with a cable same as you do now, but you will also have a wireless connection you can connect to from the DS.
Nintendo used to sell USB sticks that made a local Wi-Fi connection but they were discontinued.