The beauty of this game is that there is no "right way" to play, or "right gun" to use.
There are an insanely huge number of combinations you can utilize, & what weapons & play-style you choose is entirely up to you.
For pure reliable DPS in all situations, it's hard to beat Mercenary with a good Double Anarchy + Hellfire + Double Stinger (for longer range), but that's just one way to go.
If you use Phasewalking a lot, Catalyst can be incredibly fun, & allows for killing enemies without even needing to use guns much if you have specced into Radiance + Phoenix.
Spark is not a necessary skill per say, but what it does is increase the tech pool for elemental weapons, giving a higher chance for the element to proc (causing DoT - damage over time).
So while i wouldn't consider it mandatory early in the game, it's definitely useful later on unless you never use elemental weapons.
As skill points are limited early, i always suggest placing them based around what COM (class optimization module) you are wanting to use, as the + skill specfic boosts on the COM only apply if you have at least one point in that skill.
In terms of what one to use, it definitely is up to the individual, but as someone who's play a TON of this game, i'll mention the important ones, IMHO.
In the Controller tree, Inner Glow & Girl Power are simply amazing.
Health regen while invincible is why many call that siren ability a "god mode", overpowered, etc...
Having a boost to shield regen speed upon killing enemies makes it a lot hard to go down.
Diva is generally chosen as the route to getting down there, since Striking has limited value, especially as enemies become tougher.
Dramatic Entrance is a fun skill, but ultimately not worth putting all those points into in most cases.
Hard to Get is a must-have for any phasewalker, as it allows for phasewalking very frequently, which handy for getting around in a hurry, getting away from tough situations in a hurry, or just getting into position to attack an enemy from behind you wouldn't normally be able to, etc.
Mind Games is nice for any rapid-firing weapons as the dazing effect is more likely then, but like all dazing, i do not consider it imperative as i just don't find dazing enemies to be all that necessary.
The Elemental tree is probably the most neglected tree in the game due to it often being misunderstood & overlooked, but i consider it by far the most fun tree to spec into as a siren.
Quicksilver is an absolutely must-have 5/5 skill, zero questions about it.
Increasing firing speed for all weapons is a HUGE benefit.
Spark is very useful for elemental weapons, thought not imperative initially unless you are going down the elemental tree right away.
Resilience is largely useless sadly as not many enemies use elemental guns, & the damage reduction against isn't really enough to bother with.
Radiance is very powerful against shielded enemies & far underrated IME.
The entire key to using Radiance is playing phasewalk tag.
Basically while phasewalking, you run thru the entire group of enemies, tagging them all with shock damage from Radiance.
This removes their shields or slows shield regen extremely well, & is very helpful to any close-range phasewalking-oriented build.
You generally only need a point or two in this skill.
Venom is another sadly ineffective skill.
Melee can be fun in this game, but simply isn't very effective even with all melee skills boosted, so i just cannot recommend wasting anything more than one point in this skill (& i wouldn't even bother with that).
Intuition is by no means "needed", but it's a skill i absolutely love on a siren due to the huge speed boost gained, which allows for literally flying around enemies at ridiculous speeds
And finally, we have Phoenix, easily one of the best skills a siren can have.
First, there are drawbacks.
As Phoenix basically turns your gal into wings of fire that burn everything nearby, exploding barrels are your worst enemy, & you WILL die from damned barrels you run by until you get used to it.
Secondly, due to how it allows for ammo saving, unfortunately when the effect for ammo saving ends, you can be left with your gun not automatically reloading (which is annoying if you don't manually reload).
In spite of those drawbacks though, i still consider Phoenix extremely valuable.
The proximity fire damage it puts out is staggering, & allows you to burn enemies up easily without a single shot fired.
And the ammo-saving is incredibly useful & allows for more weapon flexibility.
If you combine a Catalyst or Firefly COM w/ Spark/Radiance/Intuition/Phoenix, you will have the ability to basically race around killing everything close range without needing to fire all that many bullets.
The Assassin tree's first line gives you Slayer, & Silent Resolve.
If you're good @ hitting the head or crit spots, Slayer is very useful.
Silent Resolve is especially helpful with phasewalking due to the damage reduction you get from exitting phasewalk.
Enforcer isn't terribly necessary unless you have a lot of points to spread around.
Hit and Run is mainly best for increasing time you can spend in Phasewalk.
I would generally recommend speccing into Slayer & Silent Resolve to unlock the the third line & not bothering with Enforcer or Hit & Run till later in the game.
High Velocity is another amazing skill you'll want at some point on every siren setup.
Increased bullet velocity is extremely helpful, & the damage boost is just icing on the cake.
Blackout is another very helpful skill for any phasewalking-heavy build.
With all the phasewalking skills maxed, especially with Catalyst, you can have a siren who can basically phasewalk almost constantly in places like Zombie Island or other places with lots of enemies.
Phasestrike is by far the most disappointing final tier skill, & simply does not work all that well, even with all melee skills maxed.
I would not bother with this skill at all.
This turned into way more of a novel than i had planned, sorry
I hope someone can gain something from the wall of text i just got carried away with...