Rules do not require an external server to run. They either execute in the SmartThings Cloud or on the hub if you are running the current beta of the hub firmware. Scenes and rules built in the mobile app are now mostly migrated to the new Rule API. You can curl the rules into the platform or generate them with postman. We have several tutorials on how to work with the new Rule API
Rule Helper Tutorial
Local Execution of Rules
If your automation does not need to connect with any third party cloud, then rules will be the most likely candidate for building automations.
If you want something more complicated or need to send/recieve data from a third party service, then Endpoint SmartApps might be the answer for you.
These are the apps that you will need to either host in your own cloud or on a raspberry pi. Here are some tutorials for that.
Developer Newsletter February 2021
Simple SmartApp Tutorial (SmartApp SDK)