With block ciphers, I understand that the plaintext is first broken down into the right-sized blocks, padded, and then encrypted into the same number of cipher blocks.
Basically, you are right. I just mention two things in addition to @poncho's answer:
(1) just as @poncho said that, block ciphers can run in different modes of operation. The message must be padded to a multiple of the cipher block size with ECB, CBC and PCBC modes.
(2) the number of padded blocks might be greater than the clear text length divides the block size. (For example, if PCKS#5 padding is used and the length of the clear text is exactly a multiple of the block size, an addition block then will be padded to the clear text.)
Once these blocks are sent over to the receiver, does this mean that (after decryption) the receiver would have to parse each block looking for the padding in order to be able to tell that the current message has ended and that the next cipher block belongs to the next message?
I do not know exactly what is "the next message" in your question; messages can be sent via different ways at different layers (@poncho uses TLS as a good example).
But whatever a message is sent, there is a way to exchange the message length between the sender and the receiver. (Also, I assume that there is a way to exchange how the cipher text is encrypted.)
So, the receiver gets the entire cipher text (by its length), decrypts it, and removes the pad from the last block (if with ECB, CBC or PCBC mode and PKCS#5 padding).
The sender and the receiver must use the other way(s) to exchange the message length. A block ends with "0x01", "0x02 0x02", "0x03 0x03 0x03" does not guarantee this block is the last block with a PCKS#5 padding; it might be right in the middle of the clear text.
Is there a way to mark somewhere how many cipher blocks a given message is, so the receiver would not have to do this parsing on every single cipher block that it receives?
@poncho mentions some ways. But whatever it is, do not depend on the padding alone to tell the end of the clear text.