New swimmer frustrated. I can only go 100m and I'm exhausted.. What gives?

Swimming can be frustrating like that.  Did this instructor observe your stroke for a while before trying to suggest these changes?  If he/she is any good they did observations and are suggesting drills based on the weaknesses they see.  3 lessons is not huge....and the fee is the going rate in my area as well.  If you want to pay less you would have to be willing to do group lessons and then you don't get as much specific advice for the flaws in your own stroke. 

Size is not always the indicator of swimming and looking good.  Some people have a natural affinity to the swim and others have had lessons or swam competitively since they were old enough to toddle into peewee lessons.  Learning proper swim form as adult takes time and patience.

That 100m is hard for you now. Make it easier by moving your threshold.

Go in tomorrow and make it 150m. Then a day or two later make it 200.  Keep adding lengths.  You'll be doing a mile or more in no time.

This is my usual advice, and it assumes you are using proper form.  If  however you are just thrashing and splashing, you will never get further without wearing yourself out.  Technique matters, you can NOT just fight the water.

Welcome to the Club, Know that you are NOT alone.. I keep praying that what everyone says is really true and if I keep getting in the water it Will get better eventually but progress is painfully slow. I even worked in another swim day so I am up to 4 times a week now.

I get the big overweight guy, and the really old guy, over in the next lane being the eveready energizer bunny's as well , so I know exactally where you are comming from there.

I was in the same boat as you a year ago except I could only do 50m before I was gassed. Now I can go over a mile and feel pretty good afterwards. The previous posts are all great tips, I started by adding 50m each workout and I slowly built my way up that way.

keep working at it and you'll be swimming miles in no time. good luck!

And, you don't think it makes sense for the OP to continue doing drills to learn good balance and streamlining first to prevent his legs from dropping?

Swimming well is about body position and streamlining first.  After that, other skills are added.  What you're describing is like trying to build a house by starting with the second floor instead of the foundation.

I'm in the learning to swim by doing drills boat too. Normally what I do, so that I'm not just doing drills the entire time I'm in the pool, is do 25m drill, 25 swim, repeat with different drills and then add one or 2 sets of longer swim lengths at the end. That way it breaks up the drills with some full stroke swimming, and the swims are short enough that I can concentrate/maintain form without getting too tired

What worked for me was learning to swim as slow as possible.  That's right, make an effort to swim as slow as you possibly can.  What this will do is teach your brain that you will indeed make forward progress (and much faster than you think) without exerting much effort.

Your brain has a natural tendancy to panic when you get tired in the water.  Your brain is saying that in order to prevent drowning, you need to stroke and kick harder.  This tends to create a snowball effect until you're basically thrashing around so bad that you come to the wall completely out of breath.

If you can convince your brain that you're not going to drown, and that simply stroking easy with a minimal kick will get you to where you want to go, then your brain will calm down, and not send messages to your arms and legs to kick it into high gear and wear yourself out.

If breathing is the only time your legs are dropping, no, I don't think it's the best use of time and money. Even then, they'd have to be dropping pretty low to be a huge concern. I've seen people whose legs dangle at nearly a 45 degree angle while breathing that can swim hundreds of meters. If you can't swim 50 yards, you're not even safe in the pool; having good breathing form wouldn't be as high on my priority list as just being able to swim. Plus, if your posture is bad when you're not breathing, that's probably the main culprit behind why it's bad while breathing. Get your body position and stroke right and there's a high likelihood your breathing posture will improve with it. I agree that streamlining is the most important part of swimming, but for the little amount of time you spend breathing, you're better off learning how to streamline the rest of your swimming and then work on streamlining your breathing.

^^^^This, unbelievably this.^^^^

Absolutely! Get the fundamentals like how to move through the water, then work on complicated stuff.

Swimming has been the biggest frustration for me of anything athletic I have done in my life.  In my opinion there is "in shape" and there is "in swimming shape".  Out of the water I am in great shape and am very active in many sports:  snowboarding, basketball, strength training, and golf and would say I am average to above average in all.  Swimming, on the other hand, is a totally different animal.  I've seen people who outside of the pool look like they can't run a mile, bench 100 lbs, hit a golf ball, or run and dribble at the same time yet they get in the pool and swim laps around me all day long.  They are amazing.  

Last weekend the longest continuous distance I had swam in a pool was a 3x25.  Last night I did a 8x25 and probably could've done 10x25.  While I have been particular about getting the right technique, or something that might resemble it, before I push myself I'm now to the point where I have to push myself past my previous limit (trying not to create too many bad habits just to get a certain distance).  It's a matter of confidence and letting go of a fear and the mindset that it's too hard.  Ok, I did a 50 so now let's go for a 75, then 100, then 150, and then 200. The key for me is to set goals that not only push myself but that are also attainable / realistic in the timeframe I'm working with.  

Less than a week ago I was frustrated with only going 3x25.  Today I'm frustrated with only going 8x25.  I'm far from being an expert but I'm confident that if I can get there you will too.