First level of Vilverin should be doable by a fresh started character, AFAICR. Don't go past the second level, though, as that's where the spooky stuff starts to appear
That rule should apply to most places near the Imperial City, with some exceptions (trying to not give too much spoilers, but stay clear of one of the caves that are near the waterfront area.)
Also, there are some easily doable initial quests around the map, like the Fishing quest in Weye, the Goblin farm in Chorrol, the ship in Anvil (once you get an enchanted weapon), Aleron Loche quest in Bravil, the guild's initial quests. Bandit camps near the road are accesible too, maybe with some patience, some luck and help from a patrolling guard (if you're on the road, use the guards as help! run to them if you're overwhelmed... that's the reason there are so many of them!)
EDIT: Sorry, meant to say that even if the quest seems to be uninteresting and/or not worth the effort since it wouldn't give you a level, for example. The truth is that some of them give good rewards, that are very useful when starting from nothing (thinking of: Chillrend from Chorrol, Anvil quest gives an enchanted Sabre, fishing quest a ring, etc.) Of course, it'd depend on which weapons your character will prefer, but they are options worth to take into account!
Rule of thumb is, the more further away you are from civilization (mostly cities) the more dangerous a dungeon will be... most of the time you'll find that you can tackle the first level, but as soon as you get into the end of the first level, or the start of the second you'll find groups of enemies, or tougher enemies, or enemies that have tons of resistances. So, stick with areas near cities at first, and progress from there.
Good luck!
Best wishes,
~Leandro.