Flame Spark and Icy Gust are both passive abilities able to be learned at level 15. Both of them do the same thing -- when you use Blink, the default Sorceress dash, you can click the attack button. If you do, the Sorceress will unleash an elemental attack where she appears. Which element depends on the skill used. Left-clicking causes fire and Right-clicking causes ice. Ice has a wider range of attack while fire deals more damage. Both skills can be learned for dual-element casters.
Phoenix Strike is a level 19 fire attack. The Elementalist fires a three-shot spread of phoenixes forward, which travel a certain distance dealing damage to enemies in their wake. The skill can be quite powerful but scales mostly with points in the skill, as opposed to weapon attack. It's useful for igniting mobs and dealing extra fire damage.
Glacial Freeze is a level 19 ice attack. The Elementalist causes a large burst of ice centered around herself, knocking enemies away with the possibility of freezing them, with the usual ice debuffs of lowered ice resist and movement speed. It's a powerful attack that scales with both weapon damage and skill ranks, but has the caveat of being centered around the caster, meaning she has to be in the center of mobs to be most effective. It is, however, quite useful as an escape technique.
Ignite is a level 32 attack that works only on enemies already on fire. It causes enemies on fire to explode violently, dealing an immense amount of damage. It's very useful. It's also quite similar to a Priest skill.
Shatter is the ice version of the level 32 Ignite, which works on frozen enemies. It causes them to shatter and take a large amount of damage, while still being left with a decreased movement speed. It's also quite useful, but less so than Ignite. The reason is because it's easier to burn enemies than it is to freeze them.
Mana Surge is a non-elemental support passive skill that you can learn at level 35, although it requires one point in Attuned Mind from the Sorceress base tree. It's quite useful, however, as it gives you a certain percent chance to recover a percentage of the MP used to cast a skill when it hits. At level one it returns 20 percent of the mana used, while all the way at level 10 that percent increases to 70.
So that's it! With this knowledge, you should be able to plan out a build that takes advantage of both elements while focusing on one or the other. Feel free to ask any questions in the comments.