In general. Especially production and economy. I've got ethiopia and the provinces all have rubber icons but I haven't seen it increase my rubber stockpile in the production tab. Is there something I need to do, like set convoys to transport it?
USEC's explanation of production is good, but I'd add that resources like Steel and Rubber and Oil aren't something you stockpile (Fuel from Oil is the only stockpiled good). You either have them or you don't. So when you started the game, you should have had like 2 rubber, and now you should have at least 3 more from taking over Ethiopia.
Most guides say to focus heavily on civilian factories to boost the economy for militarization later, but with Italy starting at war and needing to make moves early is that still the case?
Civilian Factories you use to trade for natural resources and build stuff, like military factories or more civilian factories. Building more of them early on means you can build more military factories later, but again, USEC's advice here is good. Initially its faster to get Civilian factories through focuses in the infrastructure focus tree you mentioned because they're pretty costly to build. You should always be producing more divisions though, and to arm those divisions you need military factories. You can take on minor countries with the military you've got, but in order to fight the big boys like France and UK you'll need to be constantly producing divisions.
Oil is also something I'm concerned about. I've got 5 years of fuel left and I don't know when or if I should trade for it, or how best to acquire it, aside from gunning for Romania or Iraq or something.
Fuel is from Oil, which you can trade civilian factories for. You can also get more oil from your focus tree in Libya, and by building synthetic refineries, or by conquering countries that have it (like Romania). The industry tech page has techs for increasing the amount of fuel form oil, and increasing the amount you can stockpile. There's also buildings in the construction tab you can build to store oil.
I started with Italy too, and my first game I basically used to learn how stuff worked up to like 1940, then I abandoned to play Communist France and hold the line against Germany.