I tried searching for this but couldn't find much information, so thought of creating this post as it might be helpful for other who might be looking for it.
The Microsoft docs on these methods/APIs are fairly extensive and they give a good overview of what each method does:
https://docs.microsoft.com/en-us/javascript/api/sp-webpart-base/baseclientsidewebpart?view=sp-typescript-latest#methods
https://docs.microsoft.com/en-us/javascript/api/sp-webpart-base/basewebpart?view=sp-typescript-latest
What is missing though is the order in which these methods are fired. So this post should serve as a nice complement to the docs.
I should mention that this post only focuses on the native SPFx lifecycle methods. If you are creating an SPFx webpart with React for example, the react lifecycle is out of scope for this post as there is loads of information available already on the interwebs.
SPFx webpart method execution order:
When loading the web part on a page, the methods are fired in the following order:
1) protected onAfterDeserialize(deserializedObject: any, dataVersion: Version): TProperties;
2) protected onInit(): Promise<void>;
3) protected render(): void;
4) protected onBeforeSerialize(): void;
When the web part is removed from a page, the methods are fired in the following order:
1) protected onDispose(): void;
SPFx webpart property pane method execution order:
Opening the property pane:
1) protected loadPropertyPaneResources(): Promise<void>
2) protected getPropertyPaneConfiguration(): IPropertyPaneConfiguration;
3) protected onPropertyPaneRendered(): void;
4) protected onPropertyPaneConfigurationStart(): void;
Updating the properties in the property pane:
The SPFx property pane can be set in either the reactive mode or in a non-reactive mode:
"Reactive implies that changes made in the PropertyPane are transmitted to the web part instantly and the user can see instant updates. This helps the page creator get instant feedback and decide if they should keep the new configuration changes or not.
NonReactive implies that the configuration changes are transmitted to the web part only after "Apply" PropertyPane button is clicked."
When in reactive mode, if any property is changed, the methods are fired in the following order:
1) protected onPropertyPaneFieldChanged(propertyPath: string, oldValue: any, newValue: any): void;
2) protected render(): void;
3) protected getPropertyPaneConfiguration(): IPropertyPaneConfiguration;
4) protected onPropertyPaneRendered(): void;
5) protected onPropertyPaneConfigurationComplete(): void;
When in non-reactive mode, if any property is changed, the methods are fired in the following order:
1) protected onPropertyPaneFieldChanged(propertyPath: string, oldValue: any, newValue: any): void;
2) protected getPropertyPaneConfiguration(): IPropertyPaneConfiguration;
3) protected onPropertyPaneRendered(): void;
When in non-reactive mode, after clicking on the "Apply" button, the methods are fired in the following order:
1) protected onAfterPropertyPaneChangesApplied(): void;
2) protected render(): void;
3) protected onPropertyPaneConfigurationComplete(): void;
4) protected onPropertyPaneRendered(): void;
When the property pane is closed by clicking on the "X" button
1) protected onPropertyPaneConfigurationComplete(): void;
Hope you found the post helpful! Let me know if I might have missed any. Would love to update this post in the future.