Can someone please explain how I get a dog in Skyrim? I've looked in Riverwood but I can't figure out how to actually get the dog.
You can read this article to figure how to get one:
Basically, there are several different ways. One is to find Meeko's Shack where you can simply befriend Meeko, the second is to buy Vigilance in Markarth, and the third option is to wait for a handful of random events that can result in you getting a dog.
Find the little place called meeko's shack, a dog named meeko is around it somewhere, there is a random world encounter where you find a stray dog fighting two wolves, and outside markarth a dog named vigilance is 500 to buy.
There are 2 locations of getting a dog:
Meeko's Shack, Meeko
Travel to Morthal, then go a bit north until you see Meeko. Follow him to his house and talk to him. he'll now follow you.
Markarth Stables, Vigilance
At the Markarth Stables, there is a guy who has a dog for sale 500 gold. Buy it from him, and Vigilance will follow you.
WARNING: Your dog will go to it's exact home when you tell it to go home. It'll not stay in your house. If you have Hearthfire, and have children, they'll ask you to keep the dog and then say yes and the dog wil stay in your house.
If you go to Markarth Stables, you'll find a dog by the stables. If you talk to one of the men working at the stables, you should be able to buy the dog for 500 gold.