"I already have plans."
Translation:У меня уже есть планы.
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.
And is the order important? I mean, could the order of the words be different?
Maybe у меня есть уже планы, though it is a bit sketchy. Планы у меня уже есть would also work in a rather particular context where you either inform that you made/obtained plans (which was expected of you) or reply to a suggestion to make plans.
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.")