What do you do after you beat trainer red in heartgold?

Basically it's just like all the other games, except extended a little. In all the other games after you beat the elite four the game, for the most part, is over and you're free to wander the region and try to complete any long lasting goals until you get bored and forget about the game. It's the same with this game once you beat Red.

Actually there's one other option. you can get the kanto starters and hoenn starters. The kanto you get from professor oak and the hoenn is in saffron city.

I also heard that after the kanto and hoenn starters you go to Mr. Pokemon and get the blue or red orb. Then you go to the embedded tower and groudon (red orb) or kyogre (blue orb) will appear. Although you can only get one. Trade to get the other one then show them to prof. oak for another orb. Then go back to the embedded tower and you can get rayquaza.