Area2D has a bunch of signals. The most useful ones are 'body_enter' and 'body_exit'. They will fire whenever a body enters or exits the area, and give you a reference to the body. For this case you just need to connect body_enter to a "switch room" function in your script.
You can connect signals in 2 ways.
In the editor - select the Area2D node and go to the 'Node' panel under 'Signals'. Double click a signal or click the "Connect.." button to connect a signal.
In script - use the connect() function. I prefer this because it makes them easier to keep track of. If you connect signals in the editor you will end up with functions in your script that are never called from that script. If you connect it via script you can search the script for the function and find out where it came from.
When PCMXMS said "destroy" the previous room, he didn't mean delete it forever. Presumably you will have a scene file for each room. When you want to switch rooms, you would load() the new room's scene file, create a new instance of it, and then queue_free()("destroy") the old scene since you're not using it any more. If you walked back through the doorway to that old room, it would be loaded again, just like it is in the beginning.
One day to get a functional splash screen and main menu isn't bad if you're not used to the engine yet. Unfortunately the documentation is not great. Once you learn how to do things you'll do the same work in an hour or less.