Here is what I have found:
Use RenderAction when you do not have a model to send to the view and have a lot of html to bring back that doesn't need to be stored in a variable.
Use Action when you do not have a model to send to the view and have a little bit of text to bring back that needs to be stored in a variable.
Use RenderPartial when you have a model to send to the view and there will be a lot of html that doesn't need to be stored in a variable.
Use Partial when you have a model to send to the view and there will be a little bit of text that needs to be stored in a variable.
RenderAction and RenderPartial are faster.