Maybe your approach is wrong? If your style of play is "runnin'n'gunning", you might have problems with Bioshock. Take your time to explore things (and enjoy the fantastic scenery

). Think, and plan ahead. Especially when it comes to fighting Big Daddies. Since they never attack unprovoked, it's up to you to set the time and place for the fight. Utilize the different weapons and ammo, and don't forget the PLASMIDS! (Very easy to do in the heat of the moment if you're used to regular shooters.)
And no, it actually gets easier later in the game. You get better gear and plasmids, and more experience. From what I recall from the first time I played it, the hardest parts were the very first big splicer fight while waiting to get through to Medical, Medical itself with the first BD, and the Fisheries. Hint:
At the fisheries you'll get a fancy research camera. Use it on EVERYTHING from that point.