太郎 This is Urashima Tarou presumably
を乗せた This is talking about him riding on the back of the...
亀は、TURTLE!
海の中 "through the sea"
を泳いで swimming
いく This makes it an ongoing action
と。 It's like a "quote mark" probably just giving a sense of "continuity" or "flow" to the narrative, like "and so..."
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亀は TURTLE!
ゆらりと This just means unhurriedly, like a TURTLE!
泳ぎ Swimming again
だしました。 This is like "starts out"
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さんごの林 A forest of coral
をとうり抜けて Yeah this should be とおりぬけ, "pass through" or "come out of"
く This is the same "iku" as above, making it ongoing
と、Just a conjunction ("having done this")
すきよるような屋根 This is some kind of roof, not sure what it was in the original
と、Conjunction again (just "and" in this case)
さんごの柱 Pillars of coral
できた This indicates that the following thing is made up of the roof and the pillars
りゅうぐじょう This is 竜宮城, the name of the castle where the princess lives
の前に In front of
出ました。 Came out