How to Win
Two ways to win: Get to 20pts first. Or, kill all the other monsters. Choose one and focus on it.
Center play
Early Game
Early-game: don't lose too much health trying to stay in Tokyo. The other players will output more damage on average than you will be able to deal in return.
Don't Focus Too Much on a Damaging One Single Player
Going after the single Tokyo player with 4 players is about 1/4 as good as healing yourself.
Politics
Enemy Way Ahead? Collaborate
If an enemy is way ahead of everyone else, you may need to collaborate with the other underdogs to take them down.
Create Alliances
The King of Tokyo player may take very little damage and have no incentive to leave, if each player plays in their own self-interest. You may need to agree to work together with other players to remove them, even if you would be better off individually scoring points.
Strategies for Variants
Cultists
Rolling four dice of the same type gives you a cultist. It can be used for one extra health, energy or roll. Quite important and life saver.
Berserk
Rolling four smash dice gives you berserk in next round. It will give you an additional dice which has some options like double smash, double energy. Remember not to take health because it will remove you from berserk and additional dice will be gone until you enter berserk again by four smash dice.
Curse Cards
These can be quite an interesting addition to a game. Nothing is confirmed with curse cards and the game can change any moment. A winner might lose in a single round. Be sure to keep a check on currently active curse. Keep it or change it depending on whether it's beneficial to you.
Tokyo Tower
Roll four 1 dice while being in Tokyo to enter Tokyo Tower. Rolling it once will get you additional health. Rolling it twice while being in Tokyo will give you additional health and energy. And- rolling four 1 dice thrice while being in Tokyo will make you Win the game.