Place the potatoes, eggs and 1 tablespoon of salt in a pan and fill with water. Bring the pan to a boil and let the potatoes and eggs cook for 7-10 minutes, or until tender.
Place the carrots in a separate small pan and cover them with 2-3 inches of water. Bring the water to a boil and let the carrots cook for 3-5 minutes, or until fork tender.
Drain the carrots, potatoes and eggs. Place the eggs in an ice bath, and let the potatoes and carrots cool fully in the refrigerator. While everything cools, place the red onions in a small bowl, add the sugar and pour in the vinegar.
Once everything has cooled fully, peel and chop the eggs, dice the apples and drain the onions. Place the potatoes, eggs, carrots, eggs, apples and onions in a large bowl, then add the mayo and toss.
Stir in the chopped cilantro, lime juice, and season with salt and pepper.
Transfer the potato salad to a serving dish and chill in the refrigerator until you're ready to serve it.