This quote is found at the end of Act IV, scene one, as the play nears its climax. Falsely reassured of victory by the witches, Macbeth reacts in frustration to the news that Macduff has fled to England and escaped his grasp. Macbeth then declares that he won't let anything like that happen again. From now on, he will act immediately on his desires, without hesitation. This is what he means when he says the "firstlings of my heart shall be the firstlings of my hand."
There's irony in this speech: what has Macbeth done throughout the play but act on his desires, with ever worsening results? Arguably, he's learned nothing. The wisest course he tried to take was to reconsider his desire to murder Duncan. Acting on that desire led him down a path toward greater and greater disaster. Now, however, he decides he is going to double down and plunge into more murder.
One also has to question his bold contention that he will see "no more sights," such as the ghost of Banquo. All along, he and Lady Macbeth have been declaring their toughness, but conscience continually overtakes them despite their bravado. Why should Macbeth suddenly attain the hardness he hasn't thus far achieved? An audience wouldn't be wrong to take this speech as more evidence that Macbeth is dissolving and is an unfit ruler.
The passage about which you are asking is found at the very end of ACT IV scene 1. The Weird Sisters have just shown Macbeth a series of visions which make him feel very confident that he cannot be defeated yet also very anxious. After the sisters leave, Lennox arrives, and tells Macbeth that Macduff has fled to England.
Macbeth vows to let nothing stand in the way of his dreams/plans in the following lines:
Time, thou anticipat'st my dread exploits:
The flighty purpose never is o'ertook
Unless the deed go with it: from this moment
The very firstlings of my heart shall be
The firstlings of my hand. And even now,
To crown my thoughts with acts, be it thought and done:
The castle of Macduff I will surprise;
Seize upon Fife; give to the edge o' the sword
His wife, his babes, and all unfortunate souls
That trace him in his line. No boasting like a fool;
This deed I'll do before this purpose cool:
But no more sights!--Where are these gentlemen?
Come, bring me where they are.
What Macbeth is saying is that whatever his heart first tells him to do, no matter how heinous or bloody, his hands will follow through with the action--including the murder of Macduff's family.