During Leveling here is the progression I follow.:

1 to 10 Class and Weapon skills
10 to 15 Weapon/Armor Skills
15 to 20 Second Weapon/Passives
20 to 30 Start putting points into Crafting but mostly focus on Passives and Morphs
30 and above, put points into what you need to be successful.

I don't follow this in complete lock step as if I get a class skill I want at Level 12, I will put a point into it for sure. But, when I have a bunch of skills to apply points to, and one or two points to use, I fall back on this guide.

There is a way to get a bunch of safe skyshards at L6, but that requires meta gaming (knowing the lay of the land before you get there) and I think it is better to discover the game as it is revealed the first time through. However you can do this in a modified way. When you first come to a zone, get the 7 or 8 skyshards that are not in Delves and require no combat or just some sneaking to get. Then do the zone as normal.

It depends on how you want to play.

Definitely class skills (Start out with 1 in each category so that you can unlock other skills within each of the three)
-you don't need to put a point in each skill but just one in each category to raise the overall category.

Definitely try out weapons that benefit from any passive class or racial but try to focus on no more than two weapons.
-I personally do use weapon passives but only use the healing staff active skills based on my healing Templar who can DPS too but is more so a light armor healer

Definitely try out one armor type as those passives will prove very valuable later on.

Crafting - completely optional and up to you
Racial - optional but usually we pick a race and class to work together so the passives become important


Best of luck

You will want all 3 class trees. Always have at least one skill from each tree on you bar at all times until they hit 50 (it could be an ultimate though). The reason for this is you will probably want the passives at some point. Morph skill when available to level them up to max level.

You should have all the racial passives but you might not need them as soon as they become available (my Breton doesn't really need the extra spell resistance whilst wearing light armour because of the same passive there but it wont hurt when I have a couple of spare skill points later).

Pick one type of armour to get the passives in to start with. You should wear at least one piece of each type to level them all (don't need to put points into all of them though). My magicka characters always have all the passives in light but then some of the heavy as I will run 5 light 2 heavy. If you are using restro+destro then light armour is what you should be focussing on so get the point in that rather than any others.

Get all the passives in the weapons you have equipped. Also morph the skills when available so that you can raise the skills to max level. Remember that you only get xp for the bar you are using to remember to swap it occasional or at least swap it for handing in quests as the quest xp will be rewarded to the active bar.

Look at doing 3 crafting trees max until you feel you have more skill point than you know what to do with. As long as you are de-constructing gear you will level up woodworking, tailoring and blacksmithing as you play. Alchemy and provisioning take no time to level later on in the game (just use mats from guild stores) if you want to pick them up. Enchanting takes forever so it is worth starting early with that one.

Fighters skill line passives are nice as you will often come across mobs that they can help with. The Mage guild passives only really affect their skills so invest in them if you are going to use them. Undaunted skills are pretty situational. It's good to level them but I wouldn't go out of my way to do so. The soul passives will save you some time but I generally don't bother with them until later in the game.

I generally get weapons and armour up to 50 and then respec my skills to try something new (the vet levels get dull otherwise). So I will level up to V1 with staves, light armour and magicka attribute points but then respec to medium armour, stamina and physical weapons to level them up. If you do that then you can easily respec at end game when you get bored or if a patch brings in something new you want to try. You can max out all trees to top level but then only put points in the ones you currently want to use. My V13 dk has maxed them all but currently only has points in those skill trees he will use as a tank (I ran him as a healer until a month ago), as well as a couple of crafting trees.