Technically, the best odds of getting a shiny in HGSS is via the infamous Cute Charm Glitch. You can get an in-depth guide for how to perform this glitch here, but essentially it's caused when both your ID and SID are very low numbers, which combined with a female Cute Charm user, can get your shiny odds up to 21.34%! However, without the use of RNG manipulation, (and the likely resetting of your save file,) it is an extremely long and tedious process to perform this glitch correctly, so this will most likely not be an option for you.
The next two methods have the same odds of a shiny Pokemon, but one is more likely to be accessible than the other.
One is the Masuda Method. If you have access to a Gen IV game from a different region and language of the world, you can trade a Pokemon from that game to your game. When two Pokemon from two different regions of the world breed, the eggs hatched have a higher shiny chance than regular eggs. In Gen IV, these odds are placed at 1/1638. However, if you don't have access to a different language game, this is also not an available option for you.
In this case, your "best" chance of getting a shiny is hunting in the Game Corner. If you have enough coins to buy five Pokemon, you can quickly buy 5 Pokemon to fill your party, check if they're shiny, and then soft reset if they aren't. This gives you the same odds as Masuda Method, with 1/1638.
Another viable option is soft resetting for the starters. When viewing the three starters in the starter selection screen, they will actually show if they are shiny or not. And because you can see three Pokemon at once, this technically raises your odds to 3/8192, or 1/2731.
These are your best options for shiny hunting in HGSS.
Hope I could help! Please ask if you have more questions
