I guess I'm not understanding exactly how it works.
So where are the transactions stored? Nodes?
I was studying the mining rigs and noticed there aren't any hard drives on them.
So the mining is just computing the mathematical algorithms, and fills up nodes.
So where are the nodes stored?
Correct,
Transactions are generated by anybody, then broadcasted. The nodes store unconfirmed transactions in their mempool.
Once a transaction is stored in a block, it gets removed from the mempool (it's now a part of the blockchain, so there is no need to keep it in memory).
The mining rigs = ASIC's => you are allmost correct on this one to, modern asic's have a "driver" (basically a small rpi), this driver has a small SD card to hold the driver's os, drivers for the ASIC, mining software and a small user GUI... This disk does not store any transactions on it tough, that's why your statement is still pretty good.
You're supposed to either run your ASIC on a pool, or set up your own mining node and connect your ASIC to your own node.
As for your last question: decentralised
Everybody can run a node (and many people actually do this). If you want to solo mine, or if you operate a pool, you'll have to run a node for sure...