All that is needed for this tasty treat is three pieces of wheat, three buckets of milk, two heaps of sugar and one egg.
The wheat can be harvested from planted seeds. Seeds are often produced by tall grass by hitting it. For the seeds to grow, the dirt must first be tilted. Once the block of dirt is tilted and hydrated (it'll get a somewhat darker colour when surrounded by water), you can plant the seeds. For the crops to grow more quickly, they must have a certain amount of light. The light can be natural, but it doesn't hurt to spam some torches around your crops. Make sure not to trample your crops by walking over it. Animals and enemies can also trample your crops, so be careful. The crops are ready to be harvested when they gain a somewhat yellow colour, rather than the green they previously had. Also, you'll get back your seeds after harvesting. Yay, more farming.
For the milk, you'll first need to craft some buckets. Iron buckets are made from three pieces of iron, so you'll need at least nine pieces of iron to bake some cake. To get the milk in the buckets, simply approach a cow with a bucket and press LT when near (the HUD will actually say 'Milk'). You can use different cows, but it's also possible to get all three buckets from one unlucky cow.
Sugar is produced from sugar cane, which is often found in groups near bodies of water. No doubt you'll encounter these (usually three blocks high) green plants on your travels. Hit the lowest part of the plant to make the rest come down too. The conversion from sugar cane to sugar is 1:1, so two pieces of sugar cane will do.
The egg often proves to be the hardest ingredient to get your hands on. No surprise, it doesn't grow anywhere and actually needs to be laid by a chicken. Chickens lay eggs randomly, so when encountering a few chickens, it is wise to quickly trap the little birds (build a fence or two block-high wall around them) and just wait for one or more of them to lay an egg.
Once all the ingredients are gathered, you can craft yourself a cake using the Crafting Table. Under the food tab you'll find the cake. Just select the cake, press A and the Achievement will unlock immediately.
NOTE: Cake needs to be placed to be eaten. Once placed, you can feed from it six times. It can, however, not be picked up again, so be wise when placing it.
Also, it's not a lie.
EDIT: As proven in comments, sugar cane does NOT grow in tutorial worlds. Bake your cake in a real, new game when trying to obtain this achievement!