Good early things to pick up would probably be Terror in Venice and the reprint of Arkham Horror when that comes out soon. A second Core set would also be a good thing at some point and if you are interested in playing in tournaments you definitely want the Shifting Sands pack since that has the current set of Story cards.
You mentioned being a "Hastur player". FFG hasn't put out the Hastur box yet, although some people are guessing that it might be the next one out after the recent release of the Syndicate box. We'll probably see an announcement within the next month or so and find out if that guess is correct. But aside from that there are three other things I wanted to mention:
1. Most decks in CoC tend to be dual-faction. The tradeoff of having twice as many cards to choose from is generally a good deal in return for only a very small increase in resourcing difficulty. Mono-faction decks can work, particularly if they follow a theme that has a lot of cards which buff that theme such as Deep Ones for example, but unless they are of this type they may be less competitive than a dual faction deck.
2. Most players tend to play all the factions rather than having loyalty to any particular one. They may have favorites of course, but since the cards are sold in such a way that you get all the factions, why not use them? All the LCGs are pretty much like this, although Thrones players probably gravitate a little more to their chosen houses than the other games.
3. Finally, Hastur tends to shine best as a support faction, but they don't have a lot of beefy characters which can make it tricky to run an all-Hastur deck. Historically my advice for mono-Hastur has been not to rely too much on insanity as a strategy because you will run into bad match ups against some decks who have a lot of characters that are immune to it through Terror or Willpower. So, I typically advise putting in some take-control cards to avoid having all your eggs in that one basket. This has gotten slightly harder since the most recent FAQ update when Stygian Eye and Infernal Obsession were both added to the Restricted list - you can't do *quite* as much taking control as before, but it can still be done. My expectation is that once the Hastur box does come out (whenever that may be) that Hastur will finally be in a good position to run mono and will get a lot of interesting new tools. Also, check out the Lunatics. Just because your opponent's guys might not be able to go crazy, yours still can if you want them to and they can give you some good benefits when they do.
In all the LCGs, promos are just alternate art. They do not have different rules.