Consider your Role:
There are several good things to know when choosing a race and class. The first thing you should do when deciding that you want to be, is figure out what play style you wish to spend your time on. Once that happens you can start to eliminate choices that do not fit your play style. Any racial passives which only help by granting bonuses to leveling a weapon or armor type provide only short-term gains and will not aid you during end game play. Do not worry, your character choice can always change as you figure out what play style you really enjoy playing.
One of the most important racial bonuses people should start looking at when choosing a race, are bonuses to magicka, stamina, and health and in some cases, regeneration of those while in combat. I find that these passives are highly valuable because there is no situation in the game, including the champion system, where you can increase your overall attributes for your character no matter the situation.
*Note*: There are passives such as those in the undaunted skill line that can increase magicka, stamina, and health, based on the number of different armor types you are wearing. Also, while leveling and spending champion points, your attribute pools will increase in value naturally. Typically, melee-build characters should look for races that support stamina-based play styles where casters should look for magicka. Suggested things to try would be a Redguard (king of stamina) for those melee builds, Nightblade, Templar (for DPS or tank spec), or Dragonknight. For casters who want to do magicka-based DPS, I would highly recommend the Dunmer or the Altmer, due to the bonuses they receive to fire or destruction abilities. Common classes for those are Sorcerers, Dragonknights and Templars.
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Deciding on a Faction:
Another important decision is which faction you should play. There are three factions: the Aldmeri Dominion, Ebonheart Pact, and Daggerfall Covenant. These factions are groups of people with similar goals, causes, or philosophy for which they are fighting to be the main ruling faction for the Empire. Each alliance has a chosen leader for which the players of the community fight for every day in Cyrodiil PvP to bring their faction glory. Players will be given the choice for each character they create and the faction that they choose will determine who they will be able to group with in the game since players can only group with their own faction.
Each faction has three races, from which, you can choose to play with. However if you buy the Adventurers Pack/Explorers Pack, players will be able play any race with any faction.