While i dont play an arcanist, i do play a magic ranged DPS - meaning we share the same role action pool. Some classes prefer certain actions over others, and every player has their own preferences for how to fill in the holes. So consider this advice, but not in any way binding - if you find something works better for you then it does for me you should prioritize you.
So how do you use these skills? Go into your actions menu(P is the default keybind for keyboard/mouse), go to the role menu, and pick the one(s) you want to have available. Then drag them onto your hotbar, and they cast like any other spell. A class(arcanist) should be allowed to equip one skill per 5 levels. When you upgrade to a job(summoner/scholar - though scholar uses the healing actions not the DPS ones) you can only equip one per 10 levels(maximum 5), but your new job skills should more then make up the difference. The magic DPS role actions can be divided into 3 categories depending on the intended target - you, a party member, or an enemy. So lets look at them all in their categories.
Target: YOU
Diversion: First available at level 20. For the next 15 seconds, you generate less enemy hate with your actions. This can be used to let you go nuts in a fight while still giving the tank room to hold the enemies attention. Basically worthless while soloing, this is a must have later in your career in harder group fights. Learn to use it during your big damage bursts at some point.
Lucid Dreaming: First available at level 24. Using this action cuts your current enemy hate in half and gives you a boost to your mana regen for 21 seconds. Both abilities are helpful(about to take an enemy from the tank? cut hate in half - almost out of mana? get a boost), and this will probly find a way onto your list soon after you get it. Use this when either effect is needed, bonus if both effects help you out.
Swiftcast: First available at level 32. Your next spell is cast instantly. Did your pet die and you need a new one in a hurry? Do you need to get off one of your non-instant spells NOW or while running? Need to res the healer but dont have time to stand still? Then it looks like Swiftcast is for you.
Surecast: First available at level 44. For the next 5 seconds you cant be interrupted while casting and you negate most knockback/draw in abilities. If a boss doesnt use knockback the second part isnt as useful. If most of what you want to cast is instant, the first part isnt as useful. In the right situation this one can be a lifesaver.
Target: PARTY - Note, all these abilities are basically useless while solo.
Mana Shift: First available at level 36. Gives 20% of your maximum mana to another party member. Healer running low from resurrecting a lot of people? Bard singing a song that boosts your abilities and you want it to last a little longer? These are situations where a quick shift can help. Of course, since you are giving your mana to someone else be sure you have enough to keep yourself going before you use.
Apocatastasis: First available at level 40. Reduces target party members magic vulnerability by 20% for 10 seconds. So the person you target takes 1/5 less damage from magic attacks for the duration. If you have this slotted, you can use it to reduce the damage of any big magic attack on the target - say to make sure the healer gets through the bosses ultimate move and can keep the party going. Only targets one party member, so make sure you target correctly before activating.
Erase: First available at level 48. Removes a single damage over time effect from the target, plus gives them a small heal. Using this skill you can help your party recover from certain attacks quicker. The scope here is narrow(only DoT effects on other players that can be removed), but in the right situation it can be a lifesaver.
Target: ENEMY
Addle: First available at level 8. Lowers targets intelligence and mind by 10% for 10 seconds. The intelligence and mind stats control how effectively you use combat and defensive magic spells. So you can use Addle here to debuff the enemy before an important attack/sequence. So if you can see the enemy casting something bad, this could be used to make it hurt the group less.
Break: First available at level 12. A small attack spell with a heavy effect tacked on. It does half the damage of your Ruin spell, so you arent using it as a DPS boost. But an enemy that takes longer to run up to you is one that hits you less before it dies.
Drain: First available at level 16. A small attack spell with the added effect that you heal for some of the damage done. In my experience, the heal is like 90% of the damage, up to the limit of the enemies HP(if you do 200 damage to it for an implied heal of 180 but it only has 50 health, you only will heal for 50). At higher levels you can get heals in the 1-2k+ range from this spell. An actual healer can heal you for MUCH more, but this can prove useful while solo at certain points.
So putting it all together, at lower levels when you have as many slots as you do skills you can just use everything to get a feel for them. As you level up and eventually need to choose, a couple skills start to become must have(i for one would never do a dungeon without lucid). In the end, youll probly settle on 2-4 "always useful" skills, and then you can use your personality to decide which others you like best. And keep in mind, you can switch role abilities on the fly in between fights, so if you realize you have the wrong abilities for your situation then swap in what you really need when you have a chance. So if i left any questions unanswered(or a summoner wants to give you their specific perspective), post away. Regardless, best of luck!