Traveling from the USA, staying in Dublin then going to Glasgow,

You'll clear Irish customs in Dublin, and nowhere else, assuming you're flying from a US airport directly to Dublin. As a US citizen you do not require a visa to visit Ireland and the UK, but you will need to go through the customs and immigration procedures there. Since you're staying there, the duration of this experience isn't too critical but expect to take an hour to clear customs and immigration and pick up your checked bags. During periods of congestion, longer can be required.

Ireland the the UK have a common travel area arrangement so typically, you do not clear customs when you enter the UK from Ireland. It's possible you may be flagged as a non-EU passenger and might be asked for your passport, etc. Make sure you keep a record of your flight in case UK officials ask for evidence of your date of arrival. If you like, you can voluntarily find a customs and immigration officer on arrival in the UK, but strictly speaking this isn't necessary as long as you are merely a visitor and don't require a visa. You should be subject to the same sort of time requirements as domestic passengers need. (I'd arrive 90 minutes before your flight, just in case your airline wants to inspect your passport.)

You didn't report on your return itinerary so I won't speculate.