When does a transaction end up in the transaction pool? I have been reading about the validation of transactions. And I understand the actual method in which a transaction is composed (by inputs and outputs). And I know a transaction ends up in a transaction pool before being put in a block. 

However, I want to know what happens in between. How many nodes need to validate my transaction before it ends up in the transaction pool?
The transaction pool (or mempool, as it's usually called) is not a network-wide pool. Each node maintains its own mempool.

When a node receives your tx, it will validate and add it to its own mempool, and possibly broadcast it further on. In some cases, such as a lot of unconfirmed txs, a node's mempool may be full, in which case it will not be added.

There is no network-wide consensus required to add your transaction to the mempool, each node does so individually, regardless of how many other nodes have validated that transaction.