For multiple encryption, there was a previous question: Is multiple encryption a good idea?
As to how do you know if you decrypt it correctly -- usually you look at the output and decide if it makes sense. E.g., does it have the file format of a gif/jpeg/zip file? If it was plaintext does is it all ASCII or properly encoded UTF-8? Ignoring non-ASCII code-points, only 96 of the 256 possible values of a byte are printable. So if you decrypted garbage message with the wrong key and the message was 1000 characters long, the chance that all of them are printable ASCII letters by random chance will only be (96/256)1000 or about 1 in 10400. Even allowing UTF-8 the characters have to be in certain combinations; e.g., if the first byte is of the form
1110xxxx (where each x could be either a 1 or 0), that means the next two characters both need to be the form
10xxxxxx to be proper unicode). These sorts of things rarely happen by chance.
There's also often padding (that needs a certain form in the plaintext), checksums, or Message Authentication Codes that help distinguish garbage data from the wrong key, from actual data.
EDIT: If you encrypted a file twice with two different (symmetric) keys k1 and k2 and you tried decrypting c=E(k1, E'(k2, m)), there isn't a way you could just brute-force guess k1 by just examining the output of D(k1', c), which will appear random for any pattern. You would have to brute force, k1 and k2 simultaneously to figure out properties of m.
However, its extremely infeasible to guess a high-entropy private-key by brute force. The amount of work to brute-force a 128-bit AES key is about 2128, e.g., a billion (109) computers testing a trillion (1012) keys per second for a 1000 years would only have a 1 in 10 million chance in finding the right key. So the gain from multiple encryption in this scenario doesn't seem to be much.
On the other hand, if you had asymmetric encryption with a known public-key, you can quickly check if the guessed private key is correct -- can you encrypt a message with the known public-key and then decrypt it with your guessed private key to recover the original message?