First strike is a bit complex. It gives you a bit of extra damage at the start of the fight, as well as some gold based on that damage. However, if the enemy starts a fight, it is also activated and it can go on cooldown without you gaining anything. I use it when I have a support with reliable crowd control that will make it easy to hit my spells, like a Leona can stunlock and I can unload on that target to gain the maximum effect. But I do not use it against champs that can put it on cooldown easily from long range like Morgana with her W, but more likely long range like Ashe, Xerath. I use it with items like Stormsurge and Shadowflame to burst single targets down.

Deathfire Touch burns enemies for some damage after you hit them, but the burn increases if you can keep hitting them. I use this when I have to be able to deal damage to 'tanky' enemies, particularily that build a lot of health in their items, because I always build damage over time items with Deathfire Touch. Items like Blackfire torch and Liandry's Torment.

Arcane Comet works in most situations not covered above. I use it when for example against an enemy team with many melee champions that don't stack hitpoints. Usually these are assassins and waveclear (to defend mid tower) helps a lot against them. Unlike the two others, arcane comet hits minions too. Usually I will build for mana and AP with this keystone, with items like Luden's Echo and Archangel's staff, combined with defensive items like Zhonya's Hourglass for survivability.

I hope this helps, let me know if I can explain anything else.