Before you jump into online play, unlock and play all of the characters in practice mode first. It won't take too long, and you won't see anything unfamiliar in your first few games.
Remember that every character has different loadouts, so if you don't like a character at first then try a different loadout.
You can press tab in-game to see the stats of all of the people in the match. Use this to see how powerful your enemies are and what kind of upgrades they have before you face them.
When on Aiguillon and your team has access to the stealth orb, let the teammate that can benefit from it most use it. A Clunk can use the suprise a lot better than Genji or Leon.
Try and memorize the upgrade icons, so that you can use the shop quickly and are familiar with what your teammates/enemies have bought.
When a new character comes out, for the first few days you'll want to do mostly private matches. This way you and your friends can take turns so everybody can play the character, and you can learn how to play with/against them without worry of stomping.
When a new season starts, try and play a lot at the beginning. This allows the leaderboards to settle out in a more balanced way, so that you don't get stuck in a low league later on.
Try and memorize the drop-pod path and always collect Solar on the ground, it actually helps a lot in gaining a good amount of extra Solar when it adds up.
Never try and take on the Solar Boss unless you're sure you're safe from enemies, or they can get you in a nasty spot and get you killed easily.
When using a jump-pad, jump in mid-air to cancel the jump when enemies are chasing you. They won't expect it, giving you more time to escape.
When there's a support or healer on the enemy team, take them out first and the rest of the team will go down easily.