I just acquired AGoT Core Set and I love it! What should be my next purchase and which sets should I wait to get?

As always, this depends on which house and style you want to play. If you are planning on playing Stark, Bara, Greyjoy, or targaryen, I suggest picking up their expansion box. If you want to play martell, their 3x box won't be out for a few months. Lannister doesn't have a box yet, but they are next in line.

Other than that, only the latest 2 cycles have 3x printed of each card. The brotherhood without banners set was a pretty good one.

What do you think you will be playing? That determines everything.

Well, having never played Greyjoy the Lords of the Sea looks promising since it would add another House for us to play. Let me say that I primarily will play this game with Friends and Family for now, but may get into competitive play if the community is active enough.

Quick question... If the Houses each have a 3x 156 card box, what are in the 3x 60 card boxes? Is it all Neutral cards?

They are random assortments from all 6 houses as well as neutral cards, plots, and agendas.

Levimote said:

Just to clarify, the expansion boxes contain 165 card total (3x of 55 different cards), the newer chapter packs contain 60 cards total (3x copies of 20 different cards) and I believe the older chapter pack cycles come with 40 cards (10 singles, 10 triples).

As was said before, the way you want to go with your expansion depends on whether you would like to boost any particular house.

Generally, if you want to get started on customization, it is not a bad idea to get a second copy of the core set. You'll already be familiar with the cards, but will be able to smooth out some of the kinks of the core set decks, especially by being able to swap some plots around. Also, many of the core set cards work well in a number of decks, so when purchasing further expansions and chapter packs you'll have use of many of the duplicate cards.

If you want new shinies, Lords of Winter, Kings of the Storm and Queen of Dragons all make great additions, especially since they each come with sample decklists for decks constructed from the respective box and the core set only.

Kings of the Sea and Princes of the Sun introduce new houses, which is nice, but profit more from additional support from the chapter packs. Also, they don't come with sample decklists, so you'll be on your own when mixing with the core set.

The chapter packs come in cycles (see the product page for Game of Thrones), each generally introducing or revolving around a particular theme or mechanic (but also including general purpose cards for each house). People here will be happy to recommend certain chapter packs once you decided what houses you want to boost.