sometimes - and this is the part that boggles me the most - the internet will work when you connect through the wifi extender, but NOT when you connect directly to the modem's wifi. I assume this has something to do with IP addresses, but again, I don't really know how IP addresses work, how to change mine, or whatever.

I'll freely admit my knowledge is more theoretical than the practical, but this might be a DHCP issue? Hopefully someone more knowledgeable here can confirm, but it sounds like the kind of thing that might happen if you mistakenly have multiple DHCP servers running on the same network (DHCP servers being the bits of software on your routers that hand out IP addresses). If for some reason the extender is running its own DHCP server in addition to the main router, you have a rogue DHCP server on your hands, which can cause some of the issues you're describing. The extender should not be handling anything like this--it should just be acting as an access point, which is essentially a glorified antenna that passes all traffic to your modem.

As a concrete example, I'm using a cheap router as an additional access point to extend my Wi-Fi range and when setting it up, I have to make sure to select "Access point mode" rather than the default "Wireless router mode" to avoid the rogue DHCP server issue among others. Might want to look at the manual for the extender and take a peek at the config page to confirm the equivalent setting on your device (if applicable).