your ip is public and available to any site or person you connect to. there is no way to keep it a secret. you can ask your isp to change it but your new one will be just as easy to get. it's the address of your computer. no way to get stuff delivered if no one knows your address.
an option to stay somewhat anonymous is with a VPN. that will run your traffic through a different server so someone on the other end only sees their ip address and not yours. there are some free ones but any decent one is going to cost a few bucks a month. a quick google search will turn up a bunch of options and plenty of reviews of each one.
no different than your home address really. no way for you to function on the internet without one as it tells whatever site you access where to send the data you asked for.
used to have a little widget that would display your ip on the forum everytime you loaded the page. little guy holding a sign with your ip address on it and where you're at. not seen it in a while but i'm sure that's still out there.