best class for exploring?

I'd look into:
Bandit (it's fun while solo + unfunded)
2nd job is okay and you can just explore if you don't want to mindlessly grind the efficient maps (good skill 40+)
At 3rd job you'd farm with Band of Thieves which is really nice for mobbing.

Of course you're not as fast and as efficient as assassin, but that's fine.
Either way, given you have the same HP, even doing an Assassin / Hermit would be interesting (even without washing), of course getting to boss, if you ever want to do that, will be harder to do.
But at least playing for fun it's nice

Bandit HP = Assassin HP
only thing that differs is Bandit can have low HP because of Meso Guard, assassin can't.

 

I would do hermit. The speed boost of having flash jump is insane for exploring.

They are orders of magnitude faster than any other class

 

I think the best class for you is shadower. They're a great all around class and it was also my class when I first started on legends. Flashy skills + extremely independent. Party buffs don't help too much since we don't benefit from SI and very little from SE. Endgame bossing potential is pretty decent too considering there was a recent buff, and HP washing is not needed due to meso guard and extremely high avoidability if/when you reach late levels.

You could also fly through levels in third job with meso bomb, but it doesn't seem like that aligns with what you want out of legends.

 

Yeah they do require a bit of funding, but not unless you're planning on reaching 4th job and hp washing. End game in general is pretty expensive though so I don't think it should affect your choice too much unless you plan on getting to 4th job seriously.

I agree on shadowers being great for solo-play. Minimal to no washing necessary, extremely fast grinding through 3rd and 4th job (second only to mages), decent mobbing skills, sub-optimal but still okay single-target dps, one of the best mobility (haste + assaulter).

I will say that 3rd job is pretty rough if you don't use meso explosion all the way through (also expensive). Band of Thieves does hit six mobs when maxed and hits quickly, but its range is abysmal and only activates when you directly hit a mob (range is basically as large as a normal attack). Its still a decent skill, but doesn't compare to mobbing skills for mages, bowmans, and dragon knights in 3rd job.