In battle, you can spend elixir to summon cards from your deck. Weaker cards cost less, and stronger ones cost more. You'll want to have the strongest cards in play that you can manage, but it's important to plan ahead. Always make sure you have elixir in reserve. You won't do any good by summoning a powerful creature if you're then left with no elixir to spare. Over time, you'll see your elixir gradually refill, but don't put yourself in a position where you have to wait so long for that to happen that you're taken out of play.
Join A Clan Early
When you're in a clan, you'll find a number of helpful perks. You can compete against other members of the clan, or share strategies and techniques. It's also possible to donate cards you don't need, to help other players. They might well return the favor. If you're stuck on something, it's also easy to get advice from your fellow clan members. If you can't find a clan to join, you can always create one of your own, but that will cost precious Gold. You're better of joining someone else's group, if possible, and earning cards and experience at an accelerated rate right from the start.
Level Up All Cards
When you gain additional cards that represent a particular unit, you can combine them to gain levels. This produces valuable experience, and you'll need to do it in order to reach other online battle arenas. The cost to continue upgrading increases each time you do it, but the effort is worthwhile because it will improve your overall rank and grant access to stronger cards.
Play a Bunch
Experience is its own reward, even if it doesn't come from the game itself. Keep playing and winning trophies, which will improve your rank and allow you to play against more capable players and learn new strategies from the confrontations. Sometimes, the best way to gain more understanding is to take a beating from an opponent who has already passed your current hurdles.
The Tutorial Is Worth It
Even if you know what you're doing, take a few moments to complete the tutorial. It teaches basic skills that will set you along the right path. More importantly, though, it rewards you with treasure chests and cards upon completion. That's a quick way to gain access to the first online arena, and to gain some easy gold. You can play through the tutorial again, too, if you need even more goodies.
Conserve Elixir...