I am trying to understand chronologically what happens to a transaction from the second it is sent to the moment it is added to a block.
None. The mempool is composed by unconfirmed transactions.
If you send a valid transaction to other nodes, the first node that receives can be a miner, then it puts on his/her mempool. Or it can send it to other nodes unit it eventually reaches a miner. There's no limit for how many nodes relay it.