Don't roll your own crypto.
Use preexisting protocols and libraries whenever possible. If there hasn't been any cryptanalysis on your protocol, that makes it less secure, not more. Recommended protocols are recommended because they underwent a lot of cryptanalysis and they resisted.
If you think SSL isn't right for you, you're probably just not using it right.
Your mobile device and your server already have SSL implementations. Concentrate on using it correctly.
Your comparison of SSH and SSL shows that you don't really understand what's going on. Specifically, with SSH, there are two unrelated key pairs in play: the one for server authentication (private key on the server, sent to clients and stored there), and the one for user authentication (private key on the client, stored on the server before it can be used). (The client machine's private key can also matter but only when doing host-based authentication which is quite rare.) See What is the difference between authorized_key and known_host file for SSH? for a more detailed explanation. In the case of SSL, there is no user authentication, only authentication of the server machine (and optionally of the client machine, but that's not normally done when you browse an https website).
The server will have a private key. Your mobile device needs to have the corresponding public key to ensure that it's talking to the right server. You can include the key with your application, but make sure you're able to update the key in case you want to connect to a different server. If the server needs to authenticate the mobile device or the user of the device, you will need to have a private key on the device, or a password, or some other authentication mechanism.
Understanding these aspects is the really important part. Figure out who needs to authenticate with whom, who should trust whom and who can trust whom. This will tell you which public keys you need to copy or validate. Use SSL, and make sure you use it right: the devil is in the public-key infrastructure.
I may already have mentioned this, but: don't roll your own crypto, mmkay?