It does depend on the plant. For most (dicot) flowers, veggies, etc that you will probably be growing from seed, the plant begins uptaking nutrient within a day of germination, often before soil break. However, the roots are very fragile, and cannot handle too much yet (such as the amounts of nutrients found in fertilizers).
Usually the seed stores (aka cotyledons or seed leaves) last at least until the first set of true leaves grows out. Generally, I don't begin fertilisation until the 3 leaf stage, where I begin with frequent, 1/2-2/3 strength fertilizations. When they catch on, and begin rapid growth (often around 5-6 leaf stage), I begin full-strength fertilization.
Grasses (and other monocots) are similar, but you don't see a set of cotyledons. The seed usually remains underground. But the same goes for fertilization.
This information regards seedlings in seed starting mix, which is almost devoid of nutrients. In an outdoor bed of real topsoil, the residual nutrient levels naturally found in these soils is often enough to sustain a plant through maturity without any additional fertilization. Fertilization is only necessary if the levels in the soil are too low.
Imo, in outdoor beds, the better way is to feed the soil, which will feed the plant, whereas in many pots, the mix is more like a substrate for support, and the nutrients come directly from the fertilizer. So I often use manure/compost to feed bedded plants.