"I already have plans."

Yes, it is. Here, "есть" is quite important, because it proudly proclaims the existence of plans and lands "уже".

Though, in colloquial speech this could be saved by adding an object or day which you have plans for ("У меня планы на завтра" / "У меня на тебя большие планы"). Otherwise these "already plans" do not work together—and kind of suggest that what you say is not finished.

One of the most typical ways of expressing possession in Russian is using У + Genitive + есть. У "converts" a person into a virtual "place", where certain objects can be found. Expressing possession through existence is, actually, rather common across the globe.

For real buildings and rooms we use в (e.g. "В комнате (есть) большой ковёр"~"There is a big rug in the room" / "The room has a big rug").

"У" means something like "at." The words "у меня" together mean "I have" in the sense of possession (literally "at me"). This "у" is put in front of many words in the genitive to indicate possession like "у мальчика яблоко" ("the boy has an apple"), or "у меня немного денег" ("I have a little money.")