My current understanding is that when I send a transaction I am sending it to the ethereum network. What I don't understand about this is how exactly this happens. Let's say I'm using myetherwallet, do I send it to their node and they broadcast it to all the other nodes or does my computer broadcast it to all the nodes itself? I'm just confused as to how the actual data that my computer is sending gets to all the other nodes on the ethereum network.
MyEtherWallet has a clear infographic that explains the process in a simplified way:
To answer your question, your first assumption is correct. When you use MEW and send a transaction, MEW will send your transaction to a node in the Ethereum network, and the node will broadcast the transaction to the other nodes.
Unless you are running a node locally, your computer will not broadcast the transaction directly. You are sending it to a node. In order to interact with the Ethereum blockchain you need a full node.