Small Pokémon, like Weedle, will ride on your avatar's shoulders, while bigger Pokémon will just stand beside you. There's a fun Easter egg that makes Pikachu - who normally stands beside you - ride on your shoulder, too.
Here's how to select a Pokémon to be your Buddy:
- Tap your avatar's face icon in the lower left corner of the screen
- Tap the menu button in the lower right corner of the screen
- Tap "Buddy," and select which Pokémon you want to appear next to you
You can simply pick your favorite Pokémon to be your Buddy, but you might want to use a bit of strategy: As you walk around, you'll earn candy for the type of Pokémon you have selected as your Buddy. So, something that's a bit harder to find - and thus, harder to earn candy for - might be a wiser choice than something common like a Zubat.
Additionally, some Pokémon will earn candy more quickly than others. The folks over at The Silph Road have a full chart breaking everything down.