I am new to the game, and have the Core Set plus the Queen of Dragons expansion, since I am maining Targaryen (all my friends also have core sets and expansions), and I was wondering what to buy next? I've heard many people say another Core Set to get Targaryen duplicates, which sounds good, but it still only delays my question one stage, since after I get another Core Set, I have no idea what to get afterwards for Targaryen.

There really is no way to answer this question. The answer I always give is, build the deck that you want to play online via agotcards.org then buy the packs that you need based on what you included in the deck.

I think there are certain packs that are better than others just based on the frequency with which I play the cards in them. These are the general packs I would recommend someone pick up specifically for Targ, although the order isn't that particular (especially with the Oldtown packs at the end - if you want to build a maester deck, you'll want to get them all for the various chains):

1. Trial by Combat - Meraxes, Dragon Knight
2. Where Loyalty Lies - Daario, Jorah
3. Refugees of War - Refugees of the Plain, Gilly
4. Gates of the Citadel - At the Gates (to pull Advisor turn one)
5. A Song of Summer and A Change of Seasons - Black Raven, A Time for Ravens (for summer build)
6. Ancient Enemies - Dragon Thief, Fury of the Dragon
7. Of Snakes and Sand - Jhogo
8. Tales from the Red Keep - Dragon Pit, Syrio, The Red Keep
9. The Battle of Blackwater Bay - Rhaenys's Hill, City of Soldiers
10. City of Secrets - Dragon Skull
11. A Sword in the Darkness - The Hatchlings' Feast
12. Isle of Ravens - Linked Advisor
13. Here to Serve - Valyrian Steel Link, Search and Detain

I would also recommend that you read Twn2dn's posts here on the forums and his articles on cardgamedb.com about Targ - as he's one of the best Targ players out there. Good luck!

The second Core Set is definitely a good idea, it's pretty much the best purchase a new player can make.

I would also recommend picking up the Refugees of War chapter pack. The pack contains a bunch of 0-cost, 2-str Refugee cards; Targ's are the Refugees of the Plain, which are Mil/Pow and Dothraki. Ancient Enemies is also a good choice, the pack contains the Fury plots, one for each House. Targ's is Fury of the Dragon, which isn't one of the best Furies, but it's definitely a solid card. The Furies are on the restricted list, meaning that you can only use one card on the list in your deck; I don't think you have any other restricted cards right now, and Targ is weaker on restricted cards to use, so Fury of the Dragon is a good choice. Ancient Enemies also has Dragon Thief, an awesome Ambush card that discards non-unique attachments, and Dragon bite, which I really like running in Dragon decks.

The Brotherhood cycle of cards focus on developing the Dothraki sub-theme of the House. Summer is also very popular to run out of Targ, but the chapter packs with the ravens and the Summer agenda are currently out-of-print. Many of the most recent packs also have some good Targ cards: Ser Barristan Selmy in Tourney for the Hand; Meereen Tourney Grounds, First Snow of Winter (great for getting back your Ambush cards), and Lyanna Stark *(plays well with Burn) in On Dangerous Ground; Daario Naharis and Ser Jorah Mormont in Where Loyalty Lies; and most recently Exiled Knight, Dragon Knight, and Meraxes in Trial by Combat.

The options posted above are all good starting points. I would suggest picking up the maester chapter packs pretty early in your purchases. Maesters are great for Targ and if you find out that you don't like Targ too much, they can be put into nearly any house and still be a strong deck base.

Also, Danigral posted about looking at Twn2dn's posts about Targ, which I would strongly suggest (of course from what I have seen, he can put together a mean Targ deck too)

www.cardgamedb.com/index.php/index.html/_/articles/targ-maesters-revisited-%E2%80%93-1st-place-black-frida-r58

This is his post about the deck he took to the Black Friday tournament and why he made the choices he did. It (and the linked article within it) will give you a good look at what Targ can and can't do and why maesters work so well for them.

If you want to be really cheesy, Called by the Conclave has Dragon Lore in it (its a 1 gold, maester only attatchment with the ability "kneel Draon Lore and the attatched character to reduce the cost of the next dragon you play to 0 ) you can use it with the maesters you get in the Targaryen expansion, or you could get Forging the chain to pick up the 1 gold maester so you can play one dragon for 2 gold if you set up the cards. Another chapter pack that would be fun with this is Return of the Others, it has Balerion the Black in it, a str10, 10 gold dragon with a mean ability and keywords. with dragon lore you can slap him down for free

Those cards above are probably not a great place for a beginner but if you like playing combos you might want to give it a shot, its satisfying when it works!

The extra Core(s) are really good for consistency, but especially good for Targ. Stark, Martell and GJ dont benefit from extra copies of the Core like Targ does; Forever Burning, Flame-Kissed, LDC, Fiefdoms and Xaro's Home could all be considered Targ staples.
I'd say decide what amount of burn you plan to use and acquire those CPs and 3 Cores; if you dont need or want a lot of burn then use one of the databases to pick a different theme.