Well, if you want to be breaking in everywhere then rogue would seem the most natural fit, both because you have the tools to break in and potentially sneakiness to avoid getting caught and confronted, starting fights.
Alternatively, a druid can use their wildshape forms to explore as an animal, which at least in early access meant people didn't usually challenge us when we were where we shouldn't be (I haven't played as a druid in the full release).
And a ranger, wizard or warlock with a familiar can send them in to scout, if all you want to do is explore and not rob places.
Races don't matter as much, though smaller races such as gnomes and halflings can sometimes squeeze into places others can't. Though there are options for shrinking down even if you're larger.
So in short you have many options (I'm sure there are others I haven't mentioned) and a lot of the fun of the game is finding them.
Oh, and it's true that the game doesn't explicitly talk about alignment, but not true that morality doesn't matter or isn't recognised by the game. People will definitely respond to you differently if you're helpful and kind to them. And I still have a lot of the game to go, but at least one of my companions has (correctly) commented on how my character tends to want to do good, so the game must be tracking and judging my choices in some way.