I'm absolutely astonished at those who fly fish. When they fish, it's almost like a form of art. Anyways, I really want to get into fly fishing, but I'm a bit skeptical on where to start. The fish I'm mainly going to target will be Bass, Carp (maybe), and Trout. Anyone have any suggestions on my first fly rod and reel? I'd also like to keep a budget under $150 if possible.

Agreed. Fly casting is, fundamentally, the creation of an energy wave, and transmitting it through the medium of the line, to carry the fly to the desired location. At least that's the plan.
I suggest heading down to Bass Pro/Cabelas with your budget in mind, and telling the man or lady that you want a five or six weight beginner setup. You ought to get out of there at around $250-$300 with all the bells and whistles. Seriously though, you ought to be able to get a completely satisfactory setup for under $150.

I have a bunch of used rods and some reels, and line that I could set you up a beginner's kit on the cheap. You'll want to upgrade as you become more proficient but I can supply you with:
rod
reel
line
tippet and leader
fly box
flies
You'll need line clippers and some hemostats and you're set. Even if you don't want used stuff this is a good list to begin the process-p-

Take him up on it.

My first fly rod was a TFO Professional (discontinued now) series, 5 wt 9 ft. I am about to upgrade to a more expensive rod but I still use this one and it's a great rod. You can get a starter rod from them for about $100.

I have a tfo professional 9' 5wt, Lawson liquid reel with backing and rio gold line I can sell you for $150 I prob used the rod 5 times. I just use my sage bluegill as my go to and this just sits in the closet