You're pretty much limited to traveling by ship. Daring individuals have starting making transatlantic flights, but you're still three years away from the first east-west non-stop transatlantic crossing, and also three years away from the first regular transatlantic travel by commercial airships, which will enjoy a brief period of popularity before being replaced by commercial airliners. Ironically, the airship that would make the first commercial passenger flight, the Graf Zeppelin, would also be the one to arguably spell the end of luxury airship travel when it comes crashing to earth in a fireball in 1937. You won't really be able to take a cross-Atlantic trip on an aircraft (A seaplane) until around 1937, when Imperial Airways will start making regular scheduled flights. Having said this, if you are just spectacularly wealthy, you might be able to simply purchase one of the new flying boats, have it fitted with extra fuel tanks, and just about make it from the west coast of Canada to the closest parts of Ireland.

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Your only realistic way of getting to Europe from America, or vice versa, is sea travel. You're in luck, however, because you're still in the heyday of luxury cruises! The crossing will take you a few days, maybe a week, but you'll be crossing in style, with your own state room and attendants, or if you want to slum it for some reason, you can buy a ticket on the lower decks and just wait to get there. The Titanic went down horribly in 1912, but that's in the distant past now, so you don't really need to worry about that, and the War to End all Wars ended in 1918, so you don't need to worry about the fate of the poor Lusitania, who was torpedoed off Ireland in 1915. All in all, you're relatively safe, relatively certain to make it to your destination, and you'll be doing so in style.

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I don't know much about passports at the time so someone else will have to answer that, but I hope this helps!