#1 - I get that the Core Set is needed for getting the fundamentals, but what are these different expansions I'm seeing? They range from $15 to $30, but I'm unsure what to expect from their contents. Do they expand out specific aspects of deckbuilding? Factions?
There are two boxed expansions currently, these are the $25-30 ones:
Secrets of Arkham - I recommend this be your first purchase after the Core set, it has lots of great cards (2x each) for all 7 factions in the Core set
Silver Twilight - Adds a brand new 8th faction to the game, and has a couple cards each for the existing factions. Required if you want to play Silver Twilight, all cards are 3x.
The rest are the asylum packs, which are monthly releases. There are a TON of them by now, but you can look up what cards are in each one and pick the ones you want to buy first. They all contain roughly equal numbers of new cards for all the factions, and all cards are 3x. The ones from the Rituals of the Order, Ancient Relics, and Revelations cycles contain Silver Twilight cards, the earlier ones do not.
#2 - How often do you see expansions get released? I like setting aside some "fun money" each month to keep myself from buying too many indulgence items, and I want to know how often I should expect to buy.
Until a few days ago, asylum packs came out on a monthly basis (skipping a month here and there) and the boxed expansions were always a surprise when one turned up. However, Fantasy Flight is changing the model and the last asylum pack just got released. Going forward, they will no longer be doing monthly packs - instead they will release a boxed expansion three times a year (ie - every 4 months). This has been generally met with approval as it provides a lower cost per card to buy them in bigger batches, fewer SKU's, etc… And, the game has a big enough card pool to support it. The same idea would be awful if applied to Netrunner right now which really needs cards to flow out more quickly than that.
For your purposes, you can indeed set aside a small amount and pick up one asylum pack per month, or two, or whatever level you want to spend. All of them are in print except the Dreamlands cycle, which we're hoping will return to print in the near future - possibly in a new boxed format.
The first new boxed expansion seems to emphasize Miskatonic, but they said that future ones could be based around factions, particular themes, or whatever. No real guarantees, and we haven't even seen one example to tell what "emphasize" really means. Personally, I think there will be more cards for that faction, but I predict it won't be totally lopsided - just a minor advantage.
So starting out I'd recommend a Core, Secrets of Arkham, then start picking up some asylum packs. At some point, you'll likely want a second Core set. Opinions vary, but I generally feel you should get at least 6 asylum packs before another Core. Others like to get it earlier. The tradeoff is that the Core gives you more cards per buck but they aren't *NEW* cards you haven't seen. There are a lot of really staple cards in there though. And as Runix mentioned, go out of your way to pick up Shifting Sands since it has the latest set of Story cards. There are only 3 sets of Story cards out for the game - they're in the Core, Secrets of Arkham, and Shifting Sands. So, it's quite easy to get all of them and if you ever go to tournaments only the Shifting Sands set is currently used for tournaments.