I just got the core set and was thinking of getting one or two expansions to make 1-4 decent decks out of. Which ones should I look at? Should I just stick to getting another core set? Based on the descriptions, I'd like to try Miskatonic/Shub-Niggurath for one deck. What are some of the common Faction combinations I should look into? 

I would recommend Secrets of Arkham first, then another Core, then if you like start picking up some Asylum Packs.  The Yuggoth Contract cycle is commonly recommended as a good one for new players.

As far as faction combinations, it's pretty much anything you want - they all go together just fine.

I agree with Secrets of Arkham, but I would get the Order of the Silver Twilight expansion as well so you can try out all the factions.

GongShowHost said:

I would buy a core and a Secrets of Arkham deluxe expansion. Likely the best bang for your buck for the amount of cards and the diversity offered.

Here are a couple other articles created by Chris Long during the CCG days of Cthulhu, yet still mostly relevant:

Deck Building for Beginners

Faction combinations

 +1 for another core and Secrets

Lodge is also ok out of the box, but im still torn on this fraction. Especially sacrificing your own and cards that come back to your hand is a really weird idea thats hard to make good use of. Bouncing is simpler, but also a bit tricky.

Both core and secrets have many key cards for many fractions, so getting them x2 makes much sense for pretty much all fractions. (Probably x3 would be even better, but x2 with most cards is good enough with limited pool, and too much cards that will not get x3 in a deck)

I started this way and it gives you enough cards to do some actual two fraction deckbuilding with every fraction combination. When you play with those you'll see what those decks lack and can make better AP choices depending on what fraction mix you liked most. Balance might not be ideal, but fractions show their characteristics.