There are not, actually, that many instances of anti-semitism depicted by the play itself (in fact, unless you count calling Shylock "Jew" - of which there are several instances - it's difficult to think of a specific anti-semitic action that a character undertakes). More usually a character describes something anti-semitic that has happened before.
So, Shylock, in Act 1, Scene 3, repeatedly says that Antonio has spat upon him:
You call me,—misbeliever, cut-throat dog,
And spet upon my Jewish gaberdine,
And all for use of that which is mine own.
And here's Shylock in Act 3, Scene 1 arguing to Salerio and Solanio that Antonio treats him consistently in a racist way:
He hath disgraced me, and hindered me half a million; laughed at my losses, mocked at my gains, scorned my nation, thwarted my bargains, cooled my friends, heated mine enemies; and what's his reason? I am a Jew.
There are lots of other, similar examples. But the key point here is that Shakespeare didn't write down the stage directions that usually appear in printed versions of his texts - the words are all we know are his. Never forget that these plays are meant to be acted out; and, if you think a little about what an actor might do in saying a certain speech (look at Solario in Act 3, Scene 1) you might find reasons to accuse characters of being still more anti-semitic!
Probably the best specific example of apparent antisemitism in the play is in Act 3, Scene 1. Shylock flat out states that he believes that Antonio treats him poorly because Shylock is Jewish.
He hath disgraced me, and
hindered me half a million; laughed at my losses,
mocked at my gains, scorned my nation, thwarted my
bargains, cooled my friends, heated mine
enemies; and what's his reason? I am a Jew.
Throughout the play, it is clear to audiences that Antonio doesn't like Shylock. Antonio will often refrain from actually using Shylock's name. Instead, Antonio will simply call him "Jew," such as in the line, "Hie thee, gentle Jew."
The above line happens moments after Shylock and Antonio strike a deal with each other. There is simply no reason for Antonio not to use Shylock's name right there. The line seems to ooze sarcasm and disdain about Shylock and his religion. A few lines before completely agreeing to the deal, Antonio says the following lines.
Content, i' faith: I'll seal to such a bond
And say there is much kindness in the Jew.
The line is not meant as a great compliment. Antonio is more or less stating that the only way he would consider saying that Shylock, or any Jew, as kind is if the two of them actually agree to the deal.
Most of the antisemitism in the play is more general. Shylock is the villain, and he also happens to be Jewish. What's up for debate is whether or not being Jewish is what truly makes Shylock the bad guy. I've read commentaries on this play that stress that Shylock is more of the bad guy because of his money lending habits (though this is also an antisemitic stereotype). He charges high interest rates. This means that Shylock is hated more for what he does than what his ethnic background happens to be.
I would be remiss if I didn't point out that Shakespeare makes a profound effort to humanize Shylock and make us think twice about hating him for his Jewish background. It is Shylock that points out his humanity to audiences as well as the fact that Christians are the ones who should be resented in an equal manner.
Hath not a Jew eyes? Hath not a Jew hands, organs, dimensions, senses, affections, passions? Fed with the same food, hurt with the same weapons, subject to the same diseases, healed by the same means, warmed and cooled by the same winter and summer as a Christian is? If you prick us, do we not bleed? If you tickle us, do we not laugh? If you poison us, do we not die? And if you wrong us, shall we not revenge? If we are like you in the rest, we will resemble you in that. If a Jew wrong a Christian, what is his humility? Revenge. If a Christian wrong a Jew, what should his sufferance be by Christian example? Why, revenge. The villainy you teach me I will execute—and it shall go hard but I will better the instruction.
Audiences can't help but feel sympathetic toward Shylock after this speech. If Shakespeare was truly trying to be 100% antisemitic throughout the play, I don't believe that he would have included this eloquent speech from Shylock.