The Battle Royale mode is the most popular multiplayer feature in the game, so if you downloaded the game for the sole purpose of playing Battle Royale mode, ignore everything else and complete the tutorial to unlock it, first. Don’t buy weapons or armor, as you start Battle Royale equipped with absolutely nothing. If you want to spend coins, you can buy the custom landing items (parachutes, superhero capes, etc), which are purely a cosmetic enhancement.
Winning at the Battle Royale mode is a function of evasion and escape even more than it is shooting well. Spend the initial parts of a level trying to hunt for weapons and armor, which usually tend to appear inside of buildings, with some exceptions. Until you are armed, stay away from other players, and even when you are armed, keep a distance for awhile, let them kill each other until the number drops significantly, and stay inside of the safe zone.
There are no vehicles or building capabilities in Battle Royale mode, so offense and defense are based purely on shooting and running. However, you CAN still jump just like you can in other game modes, and a high vantage point allows you to see far more of the level from where you are standing. Use jumps and hill-climbing to get to higher vantage points, look around for other players, and if you have a long-distance weapon, pick them off from the hill. Keep an escape at your back, though, just in case things go south, so that you can drop down and avoid shots as needed.
Your base is not only a chance to be creative, but literally EVERY item that you can possibly buy has a use; for example, bases give you a chance to ignore damage, and gates give a chance of a higher rocket jump (each category of item has its own effect). The strength of the effect differs based on your chosen item, though, and the effect doesn’t always follow the price. Read the item description and look for either a “+” or a “++” next to it. The “++ signifies the strongest boost, while the “+” signifies a medium-grade boost.
Be sure to check out the devices inside of the base shop, as these can significantly increase your currency earnings. Buy a Treasury and you will be able to drill out 3 coins per day. Buy a Driller to earn one gem every day. A Lucky Clover will give you one daily chance to open a lucky chest. A Jukebox will allow you to pick more different kinds of lobby music in settings. Amplifiers will incubate your eggs twice as quickly.
You earn eggs fairly frequently in this game; go to the armory in order to find them. Hit the Pets tab, then at the tabs up top, go to Eggs and tap one to hatch it once it’s ready. Once they hatch, they will move over to the Bestiary. Pets will attack enemies (or other players) for you, and you can spend coins and training to train them and level them up, which will increase their statistics.