Euro, pound, kilometres, miles
The land border is open in ordinary travel. There is no booth on most roads. The change still arrives: a speed sign in mph, a fuel price in pence, a shop that wants sterling.
Ireland and the United Kingdom operate a Common Travel Area. That is a real arrangement. It does not replace the rules that apply to your passport. If you need a visa for one side, check both governments before you travel. This desk will not tell you that you are fine.
In the car
Hire agreements sometimes limit travel into the other jurisdiction, or they charge for it. That is a contract question. Ask the desk that issued the keys. Mobile coverage and road signs change. Your map app should be downloaded for both sides before you lose a mast on a bog road.
On foot in a town
Pay in the local currency. A card often converts for you. A handful of euro coins will not help in a Derry café, and a handful of pound coins will sit in your pocket in Galway.