Here's a short list (as you can see, a complete list would be impossible) of these adverbs, or conjunctions, or
<insert POS here>s. It isn't really important how they're named; nomenclature varies from person to person like handwriting or vowel length. And there is no definitive list of Parts of Speech, anyway. What's important is how these thingies work.
also, anyway, as we were saying, consequently, finally, first, fortunately, furthermore, hence, hopefully, however, ideally, in spite of this, incidentally, indeed, instead, likewise, meanwhile, moreover, nevertheless, nonetheless, next, now, on the other hand, otherwise, regrettably, second, similarly, so, still, surprisingly, thankfully, then, therefore, third, thus, unfortunately, well, wherever we go,
Essentially, these are formulaic linking phrases (or clauses, or clauses reduced to phrases), which indicate how two clauses fit together in a context. Mostly those are, as you point out, independent clauses. However, independent clauses normally aren't marked especially. Ordinary sentences, for instance, are all independent clauses, but they have no "independent marker"; it's taken for granted, or -- as linguists call it -- unmarked in the syntax.
Normally it's the dependent clauses that need to be marked, lest they be confused with the unmarked independent ones. And also so that one has at least a prayer of discerning in which of the myriad possible ways the dependent clause "depends on" some other clause. There are a lot of kinds of dependent clauses, like complements and relative clauses, which have quite a lot of marking; and adverbial clauses, which normally use these thingies, too, though with somewhat different syntax.
Some examples of thingies that mark a clause as dependent, and indicate the manner of its dependency include:
after although how if once since lest because before than though till unless until when whenever where wherever whether while why
Notice no commas between. Generally, these are integrated into the clause syntax, rather than sitting out if front with a comma parenthesis like one finds with the linkers above. I've put all the single words above. There are a lot more thingies, but they're phrases. Some examples (separated by | because they usually don't use commas) include:
as if | as though | as long as | as much as | as soon as | as far as | inasmuch as | insofar as | even if | even though | no matter how | in that | in case | in order [that] | now that | so [that] | provided [that] | supposing [that] | given [that]
This doesn't really explain much about the various ways these thingies work, but at least there's a list or two of them. That's always a good start.