What you need to know
The Siren doesn't play like other FPS games. The Siren makes full use of Borderlands' elemental effect system and requires a more tactical level of play. Sirens eschew guns that deal the most damage at once for guns that deal elemental effects.
When you start out as a Siren, the benefits aren't obvious. You don't get the elemental guns until you're a decent level (10+). You don't start getting the good elemental guns until a bit later.
You're going to deal less damage per shot compared to other classes. It's really hard to estimate how much damage your elemental effects deal too; don't focus on the numbers, focus on the chaos you wreak. You're going to kill a lot of people, even if it's not in one shot. And in Coop play you're especially valuable; your elemental effects will distract or weaken enemies for your group to finish off easily.
Also, I hope you enjoy the sound of people burning; you're going to hear a lot of it. It's the sound of progress, my friend.
Elements
You need to know elements. Read this, the whole thing. You need to eat, breath and excrete elemental effects.
Your 30% elemental effect chance means any elemental gun with a good rate of fire is a weapon of death in your hands, even if the same gun is near useless for other classes. Experienced players will know this and often offer up elemental guns to Sirens playing Coop. Lots of people don't like/don't understand/can't use elemental guns, so you can get first dibs. This means you won't get first dibs on the high damage, non-elemental guns...but you'll quickly find as a Siren, you don't really want those non-elemental guns anyway. I keep exactly zero non-elemental guns on standby.
Also, explosive isn't an element. Not for a Siren anyway; you get no bonus out of explosive guns, give them to someone else and pick up a gun that can melt some faces.
It's possible to play Siren without focusing on elemental damage...but it just doesn't make much sense. The elemental chance percent boost is insane and costs only 5 skill points for +30% chance per shot.
Patience
Good things come to those who wait. Things like the horrible burning demise of your enemies.
Since elemental guns do Damage Over Time you'll be able to leave an enemy for dead once they've taken enough damage and enough elemental effects have stacked on them for them to die without further gunfire. Take this time to reload more often, duck behind cover to recharge shields, or light up the next target.
As you play you'll get a feel for when an enemy is "done for"; this is the point when the passive elemental Damage Over Time is likely to kill the target with no more direct attacks from you. Watch health bars closely and you'll start to get a sense for this. Don't overkill targets; move on to the next enemy. Enemies with DOT effects shoot less anyway (they're panicked) so just leave them to die and start whittling down the next target.
Note that Shock, Fire and Corrosion all have different effect lengths and guns have different damage per second; you'll need to acquire this feel per gun but you'll be amazed how quickly you start to intuit when an enemy will die. The exact times are listed here, knowing the rough length of the stacked effects will help you estimate when you can stop shooting.
Phase Lock
Phaselock has two primary purposes; to disable a dangerous target for a time so you can focus on other enemies, or to disable a target so you can easily kill it.
Note Phase Lock holds the target still, usually with their Critical point exposed and stationary. This is a great opportunity for you or your teammates to pile on the damage.
Depending on skill tree you can turn Phase Lock into an elemental Area of Effect weapon, a tool for healing yourself/teammates or a way to control enemies (or some combination thereof). They all have their uses, generally I'd pick the Phase Lock variant based on the other skills in the related tree rather than picking the tree just for the Phase Lock skills; all of them work well. Experiment and respec to try all three as you gain enough skill points
Guns
Really the Siren can effectively use any kind of gun; the reload speed, magazine size, bullet velocity and damage boost skills help all guns pretty well. From Rocket Launchers to SMGs you'll be able to hit targets more easily and spend less time reloading. You don't have (explicitly) gun specific skills unlike other classes.
Your gun choice is mostly determined by elemental effects; since your +30% elemental effect chance is per bullet, fast guns make the most sense. In the time it takes to hit one target with an epic Rocket Launcher, you could have stacked insane Fire damage with an SMG of comparable stats and left the target for dead.
Note that most non-Siren players don't like/appreciate/use elemental guns. Many will immediately pass over any Maliwan weapon because, at a glance, they do less damage. You know that's not true, but you can use this to your advantage in coop; most players will willingly give first-dibs on elemental weapons to Sirens. Just ask and make sure it's cool with your friends; you get a lot more out of those guns than they can. If you're playing with other Sirens there may still be loot fights of course.
You will seriously need one gun for each elemental type. I actually keep one SMG and one Sniper Rifle for each elemental type, but try to have at least 3 different elements equipped at once and always be ready to switch to the right element for the job. You don't always need a Slag gun, but be sure to have Fire Shock and Corrosive at the ready.
Note however that your fast reload speed, increased magazine size and reduced damage per bullet (because elemental guns do less straight-up damage) means you're going to run out of bullets. A lot. You're going to want to make sure you have at least 3 or 4 different types of guns (even if you don't keep them all equipped).
Personally I keep around a supply of Sniper Rifles and SMGs with Pistols or Assault Rifles as backups. The SMGs and Snipers run out but I'm never completely out of ammo if I keep 3-4 gun types at once.
Summary
- Use Elemental guns. Only. Keep one of each element.
- Prefer Rapid Fire guns, but other types like Snipers are okay.
- Keep backup guns. You'll run out of ammo for your favorite gun.
- Be cold, calculated and clinical. Don't overkill, let your elemental damage finish off targets.
- Don't worry about doing less damage per hit. You'll know you're doing well because enemies die, not because you deal big numbers of damage at once.
- Use Phase Lock. At the least it keeps Badass enemies out of the fight for a few seconds, at best it's an invaluable tide turning weapon.
- Don't be afraid to play support; Sirens have skills that can be great offensive and defensive support to a whole coop team
Best Answer
Fire
Fire burns Flesh; most "meaty" foes take extra damage from thus unless they resist fire (like burning psychos) or if you shoot their armor. Flesh enemies have a red health bar. Flesh enemies are extremely common, especially in the beginning of the game, so Fire is almost always helpful. Fire does much less damage to shields and armor (half or less) and many Armored foes resist Fire, so keep your fire guns away when Armored foes are about and until your target's shields are down.
Fire sets enemies on fire and causes them to panic (I'm not sure if they actually act differently yet, they did in BL1). When on fire enemies take a moderate amount of damage over a few seconds. If you note how much damage your burn effects are doing you can stop firing on an enemy once you're sure your burn Damage Over Time will kill them, making you more efficient. Remember Damage Over Time stacks so get a few good Fire effects stacked on a target before you leave them for dead.
Shock
Shock fries shields; any enemy with a blue Shield bar over their health bar will take 2x as much damage from Shock (2.5x in playthrough 2). Shock does normal damage to Flesh and Armor, so there's no need to switch guns when a target's Shield is down.
Also note that cloaked/invisible enemies can only cloak when their shields are up; additionally they all seem weak or at least vulnerable to Shock damage. Shock weapons make quick work of Hyperion Infiltrators and Stalkers.
Shock has a brief Damage Over Time effect; unlike Fire and Corrosive it does it's DOT very quickly, so concentrate fire on foes and let the DOT stack until they die. Alternately wipe out every targets' shields so your buddies can deal health damage.
Corrosive
Corrosive eats armor. Any enemy with a yellow "metal" health bar has Armor and other targets can have armored parts. Corrosive shots do less damage to health and even less damage to shields. Save corrosive guns for armored targets, but they do well enough against health in a pinch.
Targets under Corrosion are covered in green acid and the effect of their armor is weakened, which is useful in Coop when your buddies don't have corrosive guns. Corrosion lasts a long time and deals damage fairly slowly. You can stack a few Corrosion effects on multiple targets and let it eat away their health, especially effective in Coop so multiple people can take advantage of the multiple corroded targets.
Slag
Slag doesn't do extra damage on it's own, but has a chance to Slag an enemy. Purple Slagged targets take double damage from all non-slag damage for several seconds. Enemies don't appear to resist Slag as an element, but some are immune to the status effect itself. Some bosses can be slagged.
Slag is best used in coop as you can slag enemies which will take double damage from your buddies without slag damage; the more people using non-slag weapons, the more damage you can deal to a slagged target. Alternately you could use Slag grenades and non-slag guns or quickly switch to a different gun to hit a slagged target, but your weapon-switch time may eat up much of the Slag effect's length. Slag doesn't appear to "stack" in any meaningful way.
Sirens can also inflict Slag with Phase Lock via the last skill in the Cataclysm tree, or via their Melee Override Scorn in the Harmony tree. Axton can inflict slag via turrets with the Double Up skill from the Guerrilla tree.
Explosive
Explosive is the odd one out as it doesn't have a % chance of happening; explosive guns' bullets always explode and always deal a little extra damage. Explosives only deal 80% damage to shields but otherwise do normal damage to all enemies.
Explosive shots don't have a Damage Over Time effect, instead explosive bullets have a splash damage radius and always deal a little more damage per bullet than their weapon stats indicate; an explosive gun with 100 damage will hurt more than a non-elemental weapon on a direct hit, and splash damage means enemies near the bullet's impact still take damage. Multiple close together enemies can be hurt at once, too. Unlike other elements there's usually no reason to swap out Explosive guns because of an enemy's resistance, though Shields take slightly longer to take down.
As a downside to the consistently stronger damage per shot is that explosive damage can't Critical, a direct head shot from an explosive gun still isn't critical. Rocket Launchers are similarly non-critical; it's really due to splash damage more than the element, but Explosive always has splash damage. Interestingly enough, any target within the splash damage range receives full damage; it's all or nothing, and it doesn't usually matter where the shot landed on the target.
Effect Lengths
As I note in How long do elemental damage effects last?, each elemental DoT effect lasts different lengths.
So, if you're patient, a Corrosive DoT for 50 damage/sec will do much more damage than a Shock DoT for 100 damage per second.
Second Playthrough Changes
As a special note, elemental differences become more pronounced on the second pla through; damage bonuses are slightly stronger (except for Slag), and weaknesses are more pronounced. For example Corrosive will deal even more damage to Armor but be even weaker against Flesh. In your second playthrough you'll have to be even more picky about which element you use.
Damage Tables
The Borderlands 2 wiki has great damage tables for elemental effects, reproduced here with attribution: