Technically you can open the box and begin playing that day without buying anything extra but...
If you are going to have more than one person play at a time you need to buy an extra controller and an extra nunchuk. For each person who is going to play at the same time. Extra controllers "free" can be had when you buy Wii Play which is $39.99. I think you might now be able to get it "free" with Wii Sports resort too. Nunchuks cost $20/each.
You will definitely want to pick up either some extra batteries or some rechargeable ones as the ones that come in the remotes don't last long. Many recommend the recharging station for the remotes but I just use the rechargeable duracells that can also go in anything else you own.
We also have classic controllers which are good if you want to download older games using Wii Points. But again not necessary to play right away and a good item to have the kids buy with their Christmas/birthday money.
If you have an HDTV you may want to buy the upgraded cables but only if you really really care about the difference in picture quality. We have an HDTV and use the regular cables and I think the picture quality is just fine.