This is sort of a broad subject, but I've been jumping into AoS pretty hard after my 40k Hiatus (went to a two day event a couple weeks ago, and have two tournaments coming up in the next two months), so I'll try and answer these questions the best I can.
So, what are the basic tactics involved?
So, I'm going to compare to 40k, as the games have a lot of similarity, and everyone here knows the innerworkings of 40k. Also, I tend to think on a more 'competitive' instead of narrative mindset, as if you're playing narrative, nothing I say matters anyway.
So, the biggest difference in gameplay between 40k and Aos I find, is that in 40k, it's a game where you try and table your opponent, and in AOS, it's a game you play for objectives.
Tabling your opponent isn't a win condition in AoS, and it's very possible to win, and have no models left.
The best way to play AoS, is with the Generals Handbook missions (or the six in the core rulebook). They are all very tactical and well balanced on the whole. Each mission requires your army to do different things, which means your army needs to be flexible to accomplish each of the missions 'quirks'. Each 'season' of the Generals Handbook has new missions, which share themes to previous missions, but have their own quirk. 2017, was about units of 20+ models, 2018 is about artifacts and wizards.
Almost every mission features progressive scoring (in one way or another), so you need to be agressive and claim objectives eary, so you don't get too far behind on points and can't catch up.
With that out of the way, the most important things I think to have in your list right now, are screening units, and wizards (or some way to dispell spells). AoS, has a lot of units which can be incredibly fast, and Turn 1 deepstriking is a thing, unlike in 40k. taking up board space is critical, to block off where people can move (as you cannot move within 3" of enemy models, unless you're charging, or in combat), and to control where deepstrikers can go. Screens also protect your heavy hitting units from being charged too early.
With unbinding at 30", and endless spells being a thing, having a wizard is great so your opponent can't just dominate one phase of the game. Even having a cheap wizard to dispel an annoying endless spell is great to have.
What formations of units are good or useful?
This depends on the faction, to be honest. Cheap horde units to take up board space is great. Good assault units with
FLY are great top go over screens. Battalions are hit or miss, depending on the battalion. Some are fantastic, some are a complete waste of points. But, in all cases limiting your drops, the extra command point, and extra artifact is almost always worth it.
But, for Drops, either you try to go for 1/2 drops, or you just plan to not care.
How important is shooting or magic or combat compared to the other phases?
AoS is not a shooting game, so shooting is often more for support than anything else. Magic however can be quite good, depending on which wizards you have, and the spells they can cast. Every army can cast endless spells, (except like..Khorne), so no matter how good/bad your wizards warscroll spells are, you can always take those.
My Skyre Army, does a lot of damage in the hero phase with their magic. My flesh Eaters have only support spells, so they do much of their damage the old fashioned way.
Shooting is limited, as the game moves very quickly, and shooting units tend to be very expensive compared to similar units which don't. Great for trying to snipe heros, or already damage units. Artillery
can be good, but also hit or miss. If your shooting does mortal wounds though, that's pretty great. I don't think pure gunline armies are amazing competitivly, however.
I gather objective claiming is key, but what constitutes a good objective claimer?
This depends on the mission. In AoS objectives are controled by most models in 6", so, obviously hordes of models are great for that. In some missions, only heros, or wizards can control objectives. In some missions the objective moves around, or you need to take your opponents. Different units do different jobs. For my Skyre army, for example, my Verminlord goes after the objective if I need wizards/heros, as he's very fast, and has a re-rollable save, so he's tough. In other missions, I summon Giant Rats to clog up objectives.
what kind of army I should look into building.
This depends on you.
Right now, there is a difference between have (armies with battletomes), and Have not (armies without battletomes) armies.
You can also play a non-specific grand alliance force, and just go buck wild. Generic Order armies, are for example, quite strong. Chaos is also a strong 'generic' allegiance too.
I think, that all the Battletome armies are great, except for Beastclaw Raiders, and Kharadron Overlords. You
can make those armies work, but in the competitive meta, I think they lack the tools needed to compete. The others I think can all do very well.
Some factions have allegence abilties in the Generals Handbook, and I think those factions can all be strong as well.
For an allegence without either of those, you will need to work a bit harder. But, some are still quite strong. I think Spiderfang Grots arn't too bad, Order Draconis is very strong with their battation, and the Pheniox Temple has fantastic units.
If you have any specific questions, I'm happy to answer them. One of the great things about AoS, is that all the warscrolls are avalible, and all the points are avalible in the Warscroll builder page, which is fantastic. The Age of Sigmar mobile app is also very good, and if you get into playing AoS, the subscription to Azyr is very handy.
