Hatching for shiny Eggs\Eggs of a certain nature
Unlike DPPt, HGSS does not have the benefit of coin flips and happiness taps, which were used to skip through eggs until one of the desired nature and\or shininess is found. In HGSS, eggs must be hatched one at a time until the desired egg is reached. RNG Reporter (link in signature) has an option to search for very low egg PID frames, making it possible to find a shiny egg on the very first.
Hatching for flawless-IV eggs
Breeding for flawless IVs in HGSS uses very much the same procedure as that in DPPt, save for a few differences.
In DPPt, there were many wandering NPCs that made it a requirement to "calibrate" for the starting frame of an egg received from the Daycare Man. In HGSS, there is only one NPC on that route to worry about, and as a result the starting frame is usually 3 and there is no need to calibrate. To hit higher frames you have to open the Pokegear very quickly (before the NPC moves) and advance the RNG via the radio.
Furthermore, the method of IV inheritance has been changed, so all IVs have an equal change of being passed down. The Breeding (HGSS) option reflects this.
The effects of Power Items on the RNG and egg IVs is yet to be understood.
