The "long quest string" you refer to is the MSQ. They continued the story with a handful more quests every patch(usually 10-20 quests extending the story). They fall into the 2.x category, so you can do them without the expansion(though giving up the exp from them) - however they set the stage for the HW storyline(and in fact some of those quests from 3 and 4 years ago are even setting up things in the current SB storyline). Your first interaction with the Dragonsong War happens before you get into the "official" HW content. The quests introduce new people and concepts that will become important, explain why Ishgard is going to let you into their borders, and generally bridge what happens at the end of ARR with the beginning of HW. A lot of people make a big deal about the "100 quests" part. In reality most of them are quests of the run there, talk to those people, maybe kill something, report in, get your reward style - in other words fairly quick. If i remember right there are roughly 3 required dungeons during the string, and about 5 or 6 trials. And while this is subjective, most of the people ive talked to about it agree that the story starts to get really good during this quest string(trying to keep this spoiler free, so ill just say try not to skip any cutscenes). All told, even playing a casual amount this only takes a week or two to clear. Most of the quests are fairly short, so the count is inflated.

And by the way, at level 60 this all happens again as they bridge HW into SB. Only the quest string isnt as long(i think its 40 or 45) and the level 60 one has more dungeons then trials.