Big Daddies and Sisters are two most powerful enemies in the game. The first ones always accompany a Little Sister and you will have to get rid of them before adopting the Sister. Meanwhile Big Sisters appear once Daddies fail and you save/harvest each Sister on a given level.
Despite some minor similarities (like the charged attacked which you can break with the Drill Dash) and the mighty armour they both wear, the two are very different from each other and require different strategies. Let's take a closer look at them.
Since the Big Sister won't appear until all the Little ones from a given level are gone, plan before you rescue/harvest the last one - set traps and choose where you want to fight. During the fight itself, constantly move and try to guess your enemy's next move. Take advantage of the surroundings (oil, water etc.) and keep her at distance. Fighting alongside friendly Bots always makes things easier. You deal more damage by shooting the tank on her back.
Machine gun with anti-armour rounds, shotgun with phosphorus bucks or the rocket launcher with its high damage are all very useful against the Big Sister. Use Plasmids to decrease the amount of ammo used (especially since you will probably be using the more rare types). Practically every offensive Plasmid might come in handy (creating a Decoy is also useful). You could try and point out the best ones, like Electro Bolt or Winter Blast which will stun the Sister or the Insect Swarm which distract her for a while, but Telekinesis is also useful, since you can throw back items and the same goes for Hypnosis, if there are more enemies around (or you're fighting two Sisters at once).
The conclusion is easy - use whichever Plasmids you're comfortable with and try to watch your supplies. There aren't many less pleasant things than an empty clip or no EVE left in the middle of a fight.
During the fight with a Daddy, throw back his missiles and grenades with Telekinesis. Use the machine gun with anti-armour round on bigger distances and the shotgun with phosphorus bucks at closer. Don't forget about Plasmids, so that the Daddy is constantly under fire, distracted by insects, illusions or security bots. Also try to hit the tank on his back.
Rumbler is a newly introduced Big Daddy type. He throws mini-turrets and shoots rockets at you. You can approach him in two ways. You can throw back his missiles with Telekinesis and hack/throw away/destroy the turrets, but that doesn't change one crucial fact - the Rumbler series are pretty lame when it comes to melee combat. No matter the difficulty setting, if you have enough space to run around him with the Drill - do it. His only way of defence it this situation is hitting you back (which doesn't hurt that much), but if you try to constantly be behind his back (use Electro Bolt to make it easier), he shouldn't even be able to hit you too many times.