The Change Appearance portion of the alter self spell says (emphasis mine):
Change Appearance. You transform your appearance. You decide what you look like, including your height, weight, facial features, sound of your voice, hair length, coloration, and distinguishing characteristics, if any. You can make yourself appear as a member of another race, though none of your statistics change. You also can't appear as a creature of a different size than you, and your basic shapestays the same; if you're bipedal, you can't use this spell to become quadrupedal, for instance. At any time for the duration of the spell, you can use your action to change your appearance in this way again.
The key points are pretty specific, but what is allowed is very flexible as well.
It says that you can change your height and weight, but not your size. If height and weight aren't your size, how and why does this contradiction exist in the RAW?
My interpretation for this is pretty straight forward: If you are a medium sized humanoid, you can't alter yourself into a small or large sized humanoid.
A medium sized creature is anywhere from 4 ft - 8 ft tall and 60 lb - 500 lb. So alter self should work within those parameters.
As far as statistics staying the same, first thought is that it is referring to Str, Dex, Con, Int, Wis, Cha. One of that adaptations completely changes your stat block to add swimming speed and breathing underwater, adding webbing between your fingers and doing some pretty intricate changes to your lungs.
Adding flying isn't too far out of question, but how that's handled is another matter which brings me to the part about your basic shape staying the same. With the aquatic adaptation, your basic shape is the same, but now you have some pretty crazy alterations to your body.
The main thing is that you have two arms, two legs and a head. This is your basic shape. Natural alterations like claws, webbed fingers, spines, fangs, horns, are all included, but none of these are a new appendage, but instead an adaptation of something you already have. Fingers, bones, teeth, skin, etc. Giving yourself suction cup fingers would be interesting as well.
You wouldn't be able to add new wings growing out of your back or even a tail, since those are new appendages and part of your basic shape. On the other hand, I could see you changing your own arms into wings, ie; your fingers would work similarly to how a bat's would function and work out from your armpit to the end of your arm and then create a membrane. I don't think feathers would be possible since hair is a part of your basic form. If you make yourself 4 ft tall with an 8 ft wingspan, and drop your weight down to 60 lbs, I think you could fly pretty decently. You wouldn't be performing somatic gestures to spells very well though, holding any weapons at all, or be very combat oriented by any means. Whether you could add magical talons to your feet to attack with is another question all together.
As far as transforming into another race, it would have to be a race with similar basic attributes to you. You could try and make your skin look like scales or feathers, but they will just have the basic shape and colour changes you're already capable of, ie; I would ask for a deception roll to see how well you made your "scales" or "feathers" look, since touching them would give it away as being skin right away.
One last thing I can think of is darkvision. If you can breathe underwater, why can't you change how your eyes function in the dark if you give yourself cat eyes? It meets the criteria for basic shape, colouration, and distinguishing characteristics if a cats eye. The caveat is that you still need some light to distinguish details. Essentially, your low light vision would be all that improves. Imitating an eagles eyes wouldn't be out of the question either.
The tough one is echlocation. You have the basic shapes to alter; your voice and ears. Getting the use of it down might take some time and practice until you have used it a number of times and maybe even had some practice using your bat familiar to get an idea of how it works.
All of these more creative uses that aren't explicitly described in the spell should be rolled on, and perhaps even with multiple checks. Survival, history, medicine, etc. Just to get a working knowledge of what you are are actually altering. Some things are easier to do then others, so the DC could be adjusted accordingly. The nice thing is that you have an hour each time to play around with it, but unless you have some free time to play around with the spell, you aren't going to get a lot of combat uses out of it except for natural weapons which is a pretty easy concept to grasp in the first place, just like a human making themselves look like an elf would be. The DC for more difficult things should start high, and the more often you use it, slowly lower the difficulty over time until they have used it enough times to master the alteration.