DSLite? With that you can:
1) Buy the DS or DSLite browser and use it to surf the web (but TBH it isn't worth it). The DSi has an downloadable browser (like the Wii one) that is better, but still not great.
2) Play some games online against other gamers around the world. Look for the blue 'wifi' symbol on the boxes. MKDS, 42 ATC, and Pokemon are 3 games that have above average online gaming features.
3) Connect your DS to your Wii on some software / channels. Probably the most important one at the moment is the Nintendo Channel (download it from the Wii Shop Channel) which allows your Wii to transmit demos of DS games for you to play. The DS supports the ability to be used as an extra controller for Wii games but only if the software allows .. .. .. so far only Pokemon Battle thingy does this which is a missed opportunity.
What can't you do? Connect it to a PC / mac
For the online stuff your wireless network has to be either WEP security or no security, the DSLite does not support WPA.