First off I will say I am completely new to security in coding. I am currently helping a friend develop a small game (in Python) which will have a login server. I don't have much knowledge regarding security, but I know many games do have issues with this. Everything from 3rd party applications (bots) to WPE packet manipulation. Considering how small this game will be and the limited user base, I doubt we will have serious issues, but would like to try our best to limit problems. I am not sure where to start or what methods I should use, or what's worth it. For example, sending data to the server such as login name and password. 
I was told his information should be encrypted when sending, so in-case someone was viewing it (with whatever means), that they couldn't get into the account. However, if someone is able to capture the encrypted string, wouldn't this string always work since it's decrypted server side? In other words, someone could just capture the packet, reuse it, and still gain access to the account?
The main goal I am really looking for is to make sure the players are logging into the game with the client we provide, and to make sure it's 'secure' (broad, I know). I have looked around at different methods such as Public and Private Key encryption, which I am sure any hex editor could eventually find. There are many other methods that seem way over my head at the moment and leave the impression of overkill.
I realize nothing is 100% secure. I am just looking for any input or reading material (links) to accomplish the main goal stated above. Would appreciate any help, thanks.
The simple answer to how to protect the password going over the wire, replay attacks, and message tampering is: use SSL.  Yes, there are other things you can do with challenge-response authentication schemes for the login part of it, but it sounds like you really want the whole channel protected anyway.  Use SSL at the socket layer and then you don't have to do anything else complicated with how you send your credentials.
Game login authentication and security