Adding expansions almost always increases the difficulty.
And the new mechanism of Heralds (introduced in the King in Yellow) only
However, if you add more than one expansion, many of the effects that make the game harder are diluted. In fact, a board expansion and two card expansions may make the game easier than just the base game alone.
Part of the reason for this is added "timers" in the card expansions, such as the possiblility of the King in Yellow (a play being performed in Arkham in the expansion by the same name) having each of it's acts performed. If the third act is completed, everyone in the town goes insane: game over. However, if you add more cards to the deck, the chances of the cards coming up that might cause the play to progress are lower on each draw; your risk has been reduced. On the other hand, a few card mechanisms become harder to fulfill if you have diluted the deck. The Dark Pharoah expansion often requires Exhibit items, which are generally only found through events in the Dark Pharoah events.
There are solutions. Since the Mythos deck suffers from the most dilution, you can always randomly remove cards from the base Mythos deck for the cards of a second expansion (or third, and so on). Thus, for 27 King in Yellow mythos cards to go with the Dunwich Horror board expansion, you first remove 27 random cards from the basic deck. For the Dark Pharoah, you can allow all Unique item "gifts" (not the shop) to give either an Exhibit or a Unique card, your choice when you receive it.
While opinions vary, forum buzz suggests that playing with Kingsport Horror and The King in Yellow, with or without some of the basic Mythos deck is very popular. Of course the new expansion has not been out long enough to have formed a concensus, so you may want to check out The Black Goat first, if only because it is cheaper than a board expansion and readily available.