Ok I've had my iPhone 4 for just over 2 months now. In the first month I burned through my 200MB plan, so my dad switched me to the 2GB plan. Again I burned through it without noticing, and he said if I burn through my plan again then he'll take my phone and exchange it for a lesser-crap phone. I don't want that to happen, so I really need to know how to manage my wifi better.
1) What are all the things that I'll get charged for on data- like internet, youtube, etc?
2) Does using our own Wifi for our computers charge me, using someone else's Wifi, or Wifi at a Starbucks or a bookstore charge me?
3) Does the data plan reset every month meaning do I get another 2 GB to use, do I get rollover on memory like I you would for minutes for calls?
4) Do I get charged for text messages?
5) What are some other useful tips, or what are things I should keep in mind to not exceed my data plan?
2 GB is nothing, you can burn through that in just a few days. To keep a better eye on it, check your data plan by going to settings, phone and then view my data. A text message will come in to let you know how much you used. NOTE- it's not 100% accurate, but it will be a good guide. They claim that it is what is currently billed and will have to get the data from the companies that bill it to them yatta yatta. Anyway, keep an eye on it that way.
To answer your other questions. Using your own (or someone else's) Wi-Fi should not count against data that you're using. You will want to keep an eye on the icon to make sure that you are actually using Wi-Fi. When you use 3G, that's when you're burning your data. Best to keep up with your Facebook, YouTube etc when you're on a Wi-Fi. The 2G plan is not for someone who will be using the phone a lot.
You'll also want to keep an eye on what aps you have opened. Double click your button that normal gets you to the home screen. The opened aps will appear in the lower portion of the phone. Press and hold until a "-" (minus) appears. Click on the minus to close them.