Does selecting a set of random numbers for the key and IV insure security? For AES 256 - if I use a random 32 byte sequence for the 256 bit key and another 16 byte for the 128 bit initialization vector, does this insure that my cipher text cannot be broken in any short amount of time when the byte sequences are only known to me?

Related to this: how would one transmit the key to another party, so he/she may decrypt the cipher text keep it secret? Is there a preferred method, for instance by word of mouth only?
So your question has some holes in them but I will try to fill as much in as possible. If anyone were to see a mistake in my explanation, I would be more than happy if they pointed it out or edited this solution. 


AES is a block cipher that has many modes namely: ECB, CBC, OFB etc. each of these have slightly different properties (both cryptographic and non cryptographic) and benefits. That being said, AES-256 is secure (apart from ECB, and I will not dive into different different definitions of security. If interested look into CPA Secure and CCA secure). The matter now becomes which mode to use, for which there is a better explanation explained here 


Hopefully that answers your question. Let me know if there is something I missed. 


As for your second question you would want to use a public key exchange protocol. I would suggest using Diffie-Hellman Key Exchange