Long story short, I clicked a link on my phone that someone sent me. It contained an IP-logger. What information did they get from it? I'm scared because they have my IP, and god knows what other info. How bad is this and what should I do? What can they do with it?
Having your IP logged is not a big deal. Every single site you visit (and every single subresource they load, like ads) gets your IP. All your IP says is what ISP you use, and what general part of the world you are probably located in. Especially if you are using a phone your IP changes often (e.g. as soon as you switch to a new cell tower) and you probably already have a new one.
So don't worry to much about someone knowing your IP. But still, visiting shady websites isn't without risk, it's just not directly related to IP logging.
If they only have your IP address, that is not a very big deal, obviously this depends on the capabilities of that "IP-logger", if the IP-logger that you mentioned is an App that you installed, then there is a possibility that the owners can get more info about you. Otherwise, you should know that every website that you visit is logging your IP address and more information, you can search for an Apache access log if you want more details of what information a web server can save when you make a request to it. Also it is important to mention that nowadays IP addresses are dynamic, the IP address that you have this moment could change in some hours or just the next time that you turn off/turn on your router.