First, grammatically speaking, it is helpful to present the lyrics in stanzas, as in the original. The phrase you ask about starts a new stanza:
New money, suit and tie
I can read you like a magazine
Ain't it funny, rumors fly
And I know you heard about me
So hey, let's be friends
I'm dying to see how this one ends
Grab your passport and my hand
I can make the bad guys good for a weekend
The line you ask about is connected with the next one, and with this whole stanza. This stanza is talking about the boy.
To get a firmer grammatical and linguistic understanding of the first two lines, it is best to use punctuation (which bare song lyrics often do not contain):
New money, suit and tie:
I can read you like a magazine.
Linguistically the author has used topic-comment arrangement. The first part of the sentence introduces the topic, the rest of the sentence is a comment about the topic.
The first line mentions important or salient (both: "Most noticeable or important" and "Prominent; conspicuous") things about the boy:
New money, suit and tie
So the boy has "new money, suit and tie"
"New money" means that the boy has recently acquired money. "Suit and tie" means the boy is dressed nicely -- he probably bought the suit and tie with the part of the new money.
The next sentence is the comment about the topic:
I can read you like a magazine
The basic meaning of "read" is:
Look at and comprehend the meaning of (written or printed matter) by mentally interpreting the characters or symbols of which it is composed
(ODO, 1.0)
But you can apply this meaning to a person and
"Understand or interpret the nature or significance of (something)" (ODO 1.6, see also 1.5 "Discern (a fact, emotion, or quality) in someone’s eyes or expression" (or appearance).
Anyway, "read you" means "read the boy and his outward appearance," that is the 'New money, suit and tie," as these are the only things about the boy that are mentioned.
To read the boy in this way is "like reading a magazine," that is, it is very easy for the girl to do, because girls love to read magazines about style and fashion.
By the way, the song came out in 2014. This is a year after Justin Timberlake came out with a song called "Suit and Tie." Coincidence? Or is Taylor Swift alluding to Timberlake's song or to Timberlake himself? An interesting question, but not very connected with yours.