Swift uses irony in the entirety of his essay, "A Modest Proposal." Most readers probably realize that he is using irony and writing satire when he actually reveals his "modest" proposal. Swift writes that he has been assured by a "very knowing American" that a child is "...a most delicious, nourishing, and wholesome food,..." Then he announces his proposal:
I do therefore humbly offer it to public consideration, that of the hundred and twenty thousand children, already computed, twenty thousand may be reserved for breed,...[and that] the remaining hundred thousand may at a year old be offered in sale to the persons of quality, and fortune, through the kingdom, always advising the mother to let them suck plentifully in the last month, so as to render them plump, and fat for a good table.
It's possible that some readers may suspect sooner, but most will know once the lines above are read. This is certainly the point at which readers recognize the title is ironic: the speaker's proposal is anything but modest.
Let me start off by saying that the answer to your question is going to be different from reader to reader. Certain readers might pick up on the irony very early, while other readers might actually finish the essay and take the entire thing literally. That's happened in several of my English classes before. I've had students outraged that Swift could propose such a thing. That's when I begin the discussion about satire. Satire is defined as follows:
the use of humor, irony, exaggeration, or ridicule to expose and criticize people's stupidity or vices, particularly in the context of contemporary politics and other topical issues
Swift uses irony throughout the piece. Irony is defined this way:
the discrepancy between what is said and what is meant
Swift most definitely is not writing that eating children is a reasonable solution to the problems of overpopulation, starvation, and economic woes. What he is doing is calling attention to the problems that the people of Ireland are experiencing. He wants the ruling English to realize what they are doing and put reforms in place to solve the problems.
I think most readers have their "Wait, what?!" moment when Swift bluntly states his proposal.
I do therefore humbly offer it to public consideration that of the hundred and twenty thousand children already computed, twenty thousand may be reserved for breed, . . . the remaining hundred thousand may, at a year old, be offered in the sale to the persons of quality and fortune through the kingdom; always advising the mother to let them suck plentifully in the last month, so as to render them plump and fat for a good table. A child will make two dishes at an entertainment for friends; and when the family dines alone, the fore or hind quarter will make a reasonable dish, and seasoned with a little pepper or salt will be very good boiled on the fourth day, especially in winter.
Perhaps a reader would be suspicious of Swift in the previous paragraph when he tells readers about his American friend who assured him that children taste delicious, are nourishing, and can be cooked in a variety of ways.
I have been assured by a very knowing American of my acquaintance in London, that a young healthy child well nursed is at a year old a most delicious, nourishing, and wholesome food, whether stewed, roasted, baked, or boiled; and I make no doubt that it will equally serve in a fricassee or a ragout.
For other examples of irony that Swift uses, I would check the section of the text where he numbers the advantages of his proposal. My personal favorite is number six. His proposal would encourage healthier marriages and families because mothers would take better care of their children in order to bring a better market price. Fathers would take better care of their wives and never beat them for fear of miscarriage. Swift even says that the men would begin to see their wives as on par with pregnant cows.
Men would become as fond of their wives during the time of their pregnancy as they are now of their mares in foal, their cows in calf, their sows when they are ready to farrow; nor offer to beat or kick them (as is too frequent a practice) for fear of a miscarriage.