Gothenburg - North Jutland - Copenhagen

Sounds like a great plan! Going from town to town is easy by train (and bus), but a car can be useful if you want to see the countryside. Skagen is indeed beautiful and well worth a visit, it's a short train ride from Fredrikshamn. Hjørring is also easy to reach by train, but going to Blokhus will require a bus. And I'm not sure how lively it is in October.

Århus is well worth a stop on the way. And spending some time in Ålborg also seems like a good idea. Especially if your ancestors are from the northern parts of Jutland. And if you feel like a stop on Fyn, Odense is a charming place.

Thank you!

Yes, I'll also disagree with whoever claims that rail north of Aalborg is "difficult". It's neither more nor less complicated than anywhere else in Denmark (hint: it's not difficult). The public transport agency in Northern Jutland is Nordjyllands Trafikselskab, and the nationwide public transport timetable and planning resource Rejseplanen is also handy.

Also unsure of how lively it will be in October, since it's off season and Blokhus, Skagen, etc. are towns that to a large extent live of (mostly summer time) tourism. Week 42 is autumn break for Danish schools, so if your trip falls within that week you can expect a bit more life (at least for the off-season time). But do your research so you are prepared, so you don't run into a town where tourism infrastructure is closed for the off-season, and then enjoy the absence of crowds of other tourists.

Skagen is indeed a nice and famous little town. When in Skagen, I think it is also worthwhile to venture a little out of town to visit (primarily) to 'Grenen' to the north and (secondarily) Råbjerg Mile to the south. I also agree with the above recommendation of Aalborg and Aarhus, that they are worth a stop. There are also plenty of things worth seeing and doing outside the cities (depending on interests of course) - some easier to reach by public transport than others.

Grenen is lovely and a place you really should visit, but it might depend a bit on the weather. I've only been there in the summer and I'm not sure what it might be like in October, but broccolihaar probably knows more.

Skagen and the nature around it also inspired a group of artists that later became known as the Skagen painters, https://en.wikipedia.org/wiki/Skagen_Painters

I loved Aarhus. We visited friends there a few years ago and especially enjoyed the art museum, the archeology museum, DOKK-1 and the infinity bridge (it was a heat wave, so we cooled off in the water).
For "heritage" trips, I always recommend finding out as much as possible in advance so that you can see the headstones, visit the place where the family home/farm/business was/is, etc.
For Denmark, two specific recommendations. If you haven't watched Seaside Hotel on PBS, it's a lot of fun and has some gorgeous scenery shots. And I really enjoyed The Year of Living Danishly by Helen Russell, which is a narrative nonfiction book that attempts to uncover why Danes are the happiest people in the world.
Have a great trip!