Welcome to the forums, Mats
So first, if the above list doesn't make much sense to you, here is the complete names of the expensions in order: Arkham Horror (base game), Curse of the Dark Pharaoh, Dunwich Horror, King in Yellow, Kingsport Horror and just lately, Black Goat of the Woods.
I think your strategy to learning the game makes sense I wouldn't exactly buy the expansions in the order they were released, though. Dunwich and Kingsport are "bigger" expansions that actually adds a new board to the existing one, with added location and events on each. The 3 others are mainly add-on cards to the base game. Most here, I think, would agree that Dunwich Horror is one of the best expension you can buy, followed by King in Yellow. Most seems to play with any expansions they have bought, all together. It does dilutes the Mythos deck, so some players have come up with ways to alleviate that.
I personally only have Dunwich Horror and since I bought it always played it added in the base game. It does add a lot of flavor to the game!
When I solo, I started playing with just 1 investigator, but now usually plays 2 at a time. I don't think I would go higher anytime soon, mainly for space constraints on my table Still, usually, a low number of investigators usually leads to a Final Combat with the Ancient One (it's harder to close all gates or seal 6, but not unheard of...)
I think the optimal number of players for a game is 4. Having at least one of them who knows the rules could really help speed up play Arkham seems a bit more "balanced" with 4 investigators.
As a first timer in Arkham, I would suggest... to realize in advance that you will forget or misinterpret a lot of rules in your first few games but definitely after these few first game you'll set yourself a pace and rythm of play that will make your Arkham experience all the better! Also, don't be afraid to come here to ask questions (well, if you can, read the rules and the FAQ a bit, too )
And have fun!