AES-CBC then SHA vs AES-GCM for encrypting and authenticating a web token I am trying to have something like JWT but kinda ad hoc and encrypted. The token itself is simply a stringified JSON that contains the user ID and Unix timestamp.

Now, I tried to use AES-128-GCM, however I did some simple modification in the ciphertext before decrypting, just appended some bytes to the ciphertext, and found that it decrypts successfully, does that mean that those bytes were counted as padding and that AES GCM is authenticate then encrypt algorithm?

I feel that encrypt-then-authenticate feels more secure to me. Also, is AES GCM authentication even secure enough to be compared to SHA256 for example or is it CRC tier for quick integrity and cannot be used for secure authentication like HMAC?

In other words: is AES-128-CBC then SHA-256 more secure than AES-128-GCM?
just appended some bytes to the ciphertext, and found that it decrypts
  successfully


Normally this shouldn't happen, as the appended bytes surely aren't a valid authentication tag for the previous "ciphertext". I suppose (but don't know!) the implementation you are using encoded the length of the ciphertext, the associated data and the tag and retrieved these values upon decryption, ignoring the added values, yielding a correct decryption.


  AES GCM is authenticate then encrypt algorithm? because I feel that
  encrypt then authenticate feels more secure to me


AES-GCM is an authenticated encryption algorithm. Encrypt-then-Authenticate is one specific construction that achieves this general definition and is indeed preferable to Authenticate-then-Encrypt, which is why GCM internally does encrypt-then-authenticate and so AES-GCM achieves the same security definition as CBC-then-HMAC.


  is AES-128-CBC then SHA-256 more secure than AES-128-GCM?


When you say "AES-128-CBC then SHA-256" I suppose you actually mean AES-128-CBC with HMAC-SHA256 authentication on the ciphertext.

Then yes, technically that is more secure, because HMAC-SHA256 requires $2^{128}$ operations to come up with a forgery, whereas AES-128-GCM allows you to perform a multi-target search for a key that works dropping security slightly below $2^{128}$. So it also has a serious security level unlike CRC. Also from this we can infer that both are secure, as neither $2^{128}$ nor $2^{100}$ operations are feasible making them both "secure" (a.k.a. "meh, can't break"), so you might as well use the easier-to-use and faster AES-GCM, assuming you have hardware support available, which is the case on modern x86 processors.

See Squeamish's comments below for a more detailed discussion of the actual values. Also see their answer here.