Necessary Skills

Gush and Pyrogram, should definitely be leveled ASAP, and most wizards will strongly recommend Stone Rain as well (I know I recommend it).
After that, most PvE focused players will tell you to level the water shield for MP regen, and all PvP players will tell you to level the earth shield if you want to PvP.

Honestly, you can solo any non-elite (ground) mob in the game up to L60 with just gush while kiting it around - you won't even get hit. For air and water mobs, you need to add pyrogram since you can't easily kite them and I doubt you want to guzzle HP pots.

Everything else is just a nice-to-have.

Other spells that many consider worthwhile to max asap include, but are not limited to: Sandstorm, Glacial Snare, Tempest Blade, Dragon Strike, and Mountain's Seize.

They're right -- gush & pyro are basically your equivalent to a venomancer's venemous scarab. Your typical quick, effective grinding skills. Albeit you may open with stronger spells, these are usually used to kill unless the monster's element makes you have to use something else to do your best DPS. Stone rain is nice to have as well.

Tbh, leveling up your earth barrier below level 60 isn't necessary to PvP, but if you have spare SP, it's worthwhile to level it everytime you can. Personally I ditched leveling divine pyrogram to keep my earth barrier leveled and I've never regretted it.

There's nothing wrong with DP, but it doesn't become a really good skill until 59 when combined with essential sutra because of its long channeling time. If you're picking between the two to level up with your SP, level pyro rather than DP.

As for Aedril's recommendations, the last three skills listed are your ultimates (very high damaging AoEs) and you won't have the SP to max all 3 at once. You will have to pick one to max out first, and most people pick Blade Tempest because it's useful against pretty much every class. Snare and sandstorm are preferences just like DP, but sandstorm is the one you'll probably use the most in the mid levels.

@sweaty
Most people recommend that you stop leveling DP for sandstorm when you get it, though it's still up to preference. I personally love sandstorm for a lot of reasons. Shorter channel than DP (less easily interrupted), blind effect, a SS crit is huge, etc. Effectively using it depends on your stat/skill build and what kind of combo you like to work with. It's a nice skill to have for both pve and pvp; level it to 1 and play with it a bit to see what you like about it, then get a combo going with it.

Distance shrink is a skill you probably won't be leveling much for a while as level 1 will suffice and you have other priorities. As a player on the pvp server, I find it invaluable for getting away from melees before they can stun me, chasing others (especially if I slow them first), and losing other people. In pve it can be used as a secondary kite maneuver for those who do not wish to run. Very useful for soloing annoying quest bosses.