I am not sture if this information is still correct, but as far as I know bitcoin core does not work as a full node right from downloading it. I thin you have to open some ports in order for other people to connect to your node. This information can however be outdated and best is to simply follow a manual on this.
To run a node, you don't need to open any port. Connections appear to be outgoing or incoming, depending on who initiated them. For example, if we both run a node, then a connection that I have initiated between us, will be outgoing for me and incoming for you, but once the connection is established, it doesn't matter.
To allow incoming connections, yes, you must open a port, but this is definitely not an obstacle to run a node. I run a node without even bothered with ports and firewalls.
You can also use Tor and allow incoming connections over it, and you will still avoid all the networking hassle.
Finally, Bitcoin Core runs as a node instantly. After installing it and setting it to look at a specific location in your storage, then Bitcoin Core will (1) download the blockchain, (2) "talk" to other nodes transmitting information regarding blocks and transactions and (3) verify the validity of new blocks and transactions.