In a limited way.
With a fresh character, you're able to explore all zones belonging to your own alliance and Coldharbour. If you haven't unlocked the wayshrines, you'll need a player to teleport there.
In your case - Aldmeri Dominion - this limits you to the following zones:
- Khenarthi's Roost (introduction/newbie area)
- Auridon (starting area)
- Malabal Tor
- Reaper's March
- Coldharbour (don't try to do quests here before naturally progressing there to avoid spoilers and potential progression issues)
Once you've reached level 10, you'll unlock access to Cyrodiil, which is - unlike other zones -, only accessible through the campaign system (L; at the time of this writing you can still teleport to friends, group, and guild members playing there and being in your alliance, but this option is going to be removed soon due to being abused)
Finishing the guild quest lines will unlock two more small maps (with limited exploration content, but still very nice design):
Finishing the Warriors Guild permanently unlocks the Earth Forge, accessed through guild halls. Here you'll find some NPCs and two additional set crafting locations.
Finishing the Mages Guild permanently unlocks Eyevea, accessed through guild halls. Here you'll find some NPCs and two additional set crafting locations.
Once you're finished with your Main Quest (which also implies doing the story arc of your Alliance), you'll be offered to experience one of the other alliances in a what if scenario.
This won't lock you out of previous content, but essentially open up more zones in higher level variants (e.g. when visiting the Ebonheart Pact, you won't run through the same instance of the zone as starting Ebonheart players).
This will feature content starting at Veteran Rank 1 and in your case I think it would be the Ebonheart Pact, featuring the following zones:
- Bleakrock (introduction/newbie area)
- Bal Foyen (introduction/newbie area)
- Stonefalls (starting area)
- The Rift
Once done with this alliance's storyline, you'll have the option to experience the last alliance, which in your case should be the Daggerfall Covenant. The following zones will start from Veteran Rank 6:
- Stros M'Kai (introduction/newbie area)
- Bethnik (introduction/newbie area)
- Glenumbria (starting area)
- Alik'r Desert
In addition, you'll later on find the option to travel to Craglorn from any of the capitals (there's a cart waiting for you). But note that this content is meant to be played by at least 4 players starting at Veteran Rank 10 or higher.
Keep in mind that accessing some higher level zones before you progress there by playing the story, might give you spoilers of different extent on what's going to happen next.