Good morning! Sounds like a place to start and we can get into the cause. We can work out the way to test as we go along, since you do have a meter!
We don't have to get very deep into theory on RV so I often leave out small stuff on using the meter as it doesn't matter to us if we get the wrong lead on a digital meter and if you are using an older style analog meter, it will just make the meter go the wrong way and you spot that pretty quick. If your meter has a "hand" like on a clock and it goes down off the scale instead of up, change the leads to the other way! If it's a digital meter, it may tell you it is negative instead of positive but we only have one type on RV, so we can just ignore that. If were were training to be an "expert" we might fuss and want to get the black lead to the ground but I'm not fussy and just go for the handy one!
To test continuity we need to know electricity all works by a circle ( circuit?) and continuity is just making sure there is a continuous circle from one point to another. To test that you have continuity or a full circle, put the meter on the ohms scale or where there is often a funny looking upside down horseshoe marker. Touch the two leads together to get a full circle and the meter should show near zero!
They throw big words in to mess with our minds but continuity is just a circle! Your meter on ohms is testing the resistance from one point on the meter, out to the other lead and back to the meter and you have a circle!
But much of the time we need to test for voltage, either 12 volt DC or at times, 110AC, so it sounds like you have that down? Use care when around the 110AC for sure but also don't let something metal touch the 12VDC and ground at the same time! Big sparks and they can burn you!!
I looked at this drawing and down on page four:
I did a snip of the drawing and did some marking to make it a but if easier to read you want to see the whole story it on the link. If you are using a computer, click this picture to get a better view.
Down at the bottom, it shows power comes in on the cord, either from the power pole or the generator and goes to a 30 amp breaker. That 30 amp then feeds the other smaller breakers in that box. Sine you have the green stuff working, we can skip over much of the box as we know the 30 and the two 20 at the right are okay!
But we can see that the first 15 amp breaker from the left feeds a bunch of stuff as well as the converter and we don't know if there is a problem at that breaker!! If the connection at the breaker is loose or the breaker is bad, that would cause that group of outlets in the bedroom, etc and likely the converter to not work, so look at that, flip the breaker on/off and make sure it goes all the way on and look for loose wires there.
But it may be easier to look at the GFCI in the bathroom first? IF the GFCI outlet is working but not the rest of the circuit (circle?) is not, I might think it is a loose wire on the back of the GFCI outlet. when the GFCI is not working right, it should trip or cutoff both that outlet and the rest of the group! Not always true but that would be expected so some looking at the back of that outlet is needed. The wires should be marked on the outlet in some way like " line in, or field out" as it is kind of like a switch where power comes in, goes through the GFCI and then goes out to the "field" stuff on down the line.
On the house lights and things, do you know about a battery disconnect switch that is usually near the entry door? If it is off, that makes almost all the 12 VDC like lights, water pump, vent fans, etc. dead, even if you have good batteries! Maybe a problem or maybe just needed to turned on? Go for the easy stuff and hopes it just needs turned on??
Maybe see what you find and we can work on any other problems/ questions as you go.