This is a complicated and fraught topic, I know. My family have already booked travel for spring 2020 and one leg of our trip is a flight from Dublin to Glasgow. From what I can see, our US passports will be what we need, but most of the information out there is directly related to UK-EU travel (and rightly so). Does anyone have any information?
You will be fine, it almost certainly won’t affect you at all.
This topic has been done to death on here so if you search the forum you will find posts with all the information that we have, and a lot of educated guess work. There really isn’t much we can add. :-)
It may simply mean your dollar won't go as far in the future. As ever, wait and see.
My answer to the question in the title of the post is this: very little.
If they leave with a deal: There will be a transition period until 2020-12-31 during which very little will change.
If they leave without a deal: Nobody knows, but ask again in February.
Although, if Nicola Sturgeon succeeds in leading Scotland to leave the UK, things could get really interesting. Presumably, they will join the EU, thus eliminating any trade barriers between Ireland and Scotland. Curious and strange events.
Brexit will have little effect on you.
As of right now, if you travel from Dublin to Glasgow, it is an open border and you won't even need to show your passport on arrival. If there is a change, the only one would be that you would then have to go through UK immigration. And in most cases, you can use the electronic gates.
I'm in the UK now, I'm planning a return trip in the spring and it will include both the UK and some Schengen countries. I'm not worried.