Both Albus and Scorpius continue their education after Hogwarts. Scorpius studies History of Magic to become a teacher or a scholar, though I’m also keen on idea of Scorpius becoming a Healer and treating history as a hobby. Albus becomes a Magizoologist, a Potioneer or a Healer (I’m not sure which career suits him the best, though I definitely don’t see him as an Auror).
Harry and Draco rent for their sons a little house. They want Albus and Scorpius to focus on their education and not to worry about money and jobs so they pay the rent and give them money for life during the boys’ studies. Besides, what’s a better way to spend their fortunes than to make their children happy and comfortable? Harry does the same for James and Lily.
I imagine most post-Hogwarts schools are located in London or other big cities but Albus and Scorpius prefer to live in a smaller town. They like peace and quiet and an easy contact with nature. There’s a lot of meadows and forests around their town where they love to spend their free time.
They travel to schools via Floo network or Apparition. Scorpius passed his Apparition exam at the first attempt but Albus failed. James laughed at his younger brother and teased him for being so small and still having problems with Apparition. Of course, Albus passed with flying colours on the second attempt and is now better in Apparition than Scorpius who hates it because it makes him sick.
A few times a month Albus and Scorpius travel to London or other cities to meet with their friends or to go to a nice restaurant, cinema, theatre, concert, etc. They avoid big parties, discos, these kind of stuff, it’s definitely not their cup of tea. Because of Scorpius’ obsession with history, they also visit a lot of historical sites and museums, both magical and non-magical. Albus likes these visits as well, Scorpius’ stories about interesting historical events and figures are far more fascinating than professor Binns’ boring lectures.
Albus always wanted a dog. He loves long walks during which he can relax and think about many things and a dog is a perfect walking companion. He wants a loyal friend, whom he can tell about absolutely everything without fear of being judged, who will never betray his secrets. Obviously, Albus couldn’t take a dog to Hogwarts and he didn’t want a dog just for two months a year (besides Harry and Ginny warned their son that they wouldn’t have time to care for a dog during Albus’ absence) so he had to wait a long time to finally get one. When he returned home after seventh year, the best gift he’d ever gotten was waiting for him – a beautiful and happy puppy (at last Harry learned his lesson and understood that there were better presents than old blankets).
The dog is medium size; as Albus is small, he needs a dog that won’t be larger and stronger than his owner. It may be a border collie (I have one so I’m biased) or a dog which Sam Clemmet has (does anyone know what breed it is?).
Scorpius loves Albus’ dog and treats it as his own. Despite his best attempts, he wasn’t able to teach the dog not to sleep on the bed (Albus wasn’t helpful at all) and it drives him mad when he sees dog’s hair on their clean bed sheets. But he adores watching Albus interacting with his new friend, how happy and carefree he is. Scorpius especially loves secretly watching his boyfriend talking to the dog. Albus says funny things in such a silly voice when he thinks he’s alone with his dog. Once he caught Scorpius eavesdropping on him and was so mad that he didn’t speak to him for two full hours.
Contrary to popular belief, Albus and Scorpius like sport. They’re just not fond of quidditch, they prefer individual sports, with less risk of contusion. They like swimming, skiing, jogging but above all cycling. It allows them to visit countryside, spent a lot of time outdoors and, what’s important, they can chat all the time during this activity.