I would consider using TeamViewer.com to access the Mac remotely (free for personal use). It allows file transfer. LogMeIn.com also allows remote screen sharing, but you pay for file transfer.
OR, enable the iCloud "Back-to-My-Mac" feature that will also allow you to access your Mac remotely. This will allow screen sharing as well as file sharing.
You could also look at Hamachi VPN service (free for personal use). Hamachi is another service offered by LogMeIn.com
I do not have anything against ssh (I use it every day), but it is not easy to get everything configured correctly for remote across the internet access.
You will need to open ports in your home router for the Mac you want to connect with. See <http://portforward.com>. They will give you instructions for forwarding the ssh port 22 from the internet to your destination Mac. This is done on the router where your destination Mac is located.
Then you should consider getting a free dynamic DNS name so you can address the destination Mac by name instead of my an IP address that the destination Mac's ISP could change any time it feels like it. No-IP.com and DynDNS.org offer free dynamic DNS names. You will need to run a dynamic DNS updating utility on the destination Mac. The dynamic DNS name provider has these utilities, just download theirs.
Now that you have Remote Login enabled, your router doing the correct port forwarding, and a dynamic DNS name, you make your ssh connections
ssh username@dynamic.dns.name
scp username@dynamice.dns.name:/path/to/file/you/want/to/copy /local/path/to/store/file
scp /local/file/to/copy username@dynamice.dns.name:/path/to/store/file
If you succeed in doing this, then you can expand into ssh tunnels, where you can Screen Share and File Share across ssh tunnels 🙂