At the start you have almost all characters and limited courts. Here's how you unlock them.
Characters
Shy Guy and Donkey Kong Jr shouldn't be too hard to unlock. Shy Guy can be unlocked through the third part of the tournament which is the Star tournament. Donkey Kong Jr is the same but you need to play through as a double instead of a single. You can also unlock Alex, Harry, Nina and Kate through transfer pak. Follow the prompts and choose whatever style fits their character. It will be great to earn points and then send back to the GBC version as many times as you want. However if you download through Virtual Console, those four from the Game Boy Color wouldn't exist.
Courts
You'll easily grow bored playing in those average courts. There's going to be a number of ways to unlock them so you can enjoy the speed and bounce with what kind of court you'll be playing in.
Without GBC Transfer
- Donkey Kong Court can be unlocked through Mushroom Cup in Singles with Donkey Kong.
- Mario Brothers Court can be unlocked through Mushroom Cup in Singles with Mario.
- Mario Brothers Castle Court can be unlocked through Star Cup in Singles and Doubles with Mario.
- Piranha Plant Court can be unlocked if you perfectly complete with any character in Piranha Plant Challenge.
- Second Mario Brothers Court can be unlocked through all cups in doubles with Mario.
- Wario Brothers Court can be unlocked through all cups in doubles with Wario.
- Yoshi and Baby Mario Court can be unlocked through Mushroom Cup in singles with Yoshi.
- Yoshi and Birdo Court can be unlocked through Star Cup in doubles using Birdo.
With GBC Transfer
- Bowser Court can be unlocked through Bowser's minigame before transferring to the N64.
- Peach Court can be unlocked through Peache's minigame before transferring to the N64.
- Waluigi Court can be unlocked through Waluigi's minigame before transferring to the N64.
- Wario Court can be unlocked through Wario's minigame before transferring to the N64.
- Yoshi Court can be unlocked through Yoshi's minigame before transferring to the N64.
Special note: If you have the requirements of GBC Transfer Pak, Game Boy Color and the game Mario Tennis for that handheld, you just follow the instructions and you can unlock the courts and characters without any trouble. However if you don't have those requirements or you download the Virtual Console version, you're limited to what you can unlock without those wonderful extras.