House Trim

Now that spring has arrived I’ve located a few new issues with the outside of the house.  I’m looking for some help.  The thing with this house is – I’ve never done this type of stuff before and I’m just learning as I go.
I was not happy to pull in my driveway one night after work and noticed that the wood trim under the gutter of the sunroom was completely rotted out.

I’m not quite sure how to fix this.  My assumption is that I need to get the gutter off and then look at the wood and hope that the studs are not rotted.  I looked on another part of the house and the trim appears to be molding???

Was this a trend back in the 1920’s?  Was this an update to the house?  So should I use molding or just some treated wood?  Anyone’s advice would be helpful.