I hate being the bringer of bad news, but if you're that new to programming, making a secure networking app is not your best shot, unless you can get an experienced programmer to guide you the whole way.
About sockets:
http://compnetworking.about.com/od/itin ... 83100a.htm
VPS (This is simplified quite a bit) means Virtual Private Server.
The "Virtual" part is because you share hardware with other VPS instances.
Each instance is as far as it knows, its own machine.
Running a VPS is pretty much the same as running your own machine on the internet, and connecting to it through the network/internet to maintain and set it up.
So a VPS just lets you run an operating system, nothing more.
You can run Linux and Windows on most VPS systems, Linux the most common but Windows a bit easier for a beginner but costs a bit more per month.
As for encryption, I'm not a cryptological expert but I do use it in my work as a professional programmer.
If you're encrypting the data before sending it and using secure keys, you don't need to encrypt the traffic in other ways, unless you're really paranoid or handling VERY sensitive data.
Dealing with the data on the server has nothing to do with the VPS, remember that's just a computer on the 'net, but has everything to do with the server software.
The server software takes the data, decrypts it, and acts on the information.
My honest opinion is this:
Learn how to do it locally first, make a few "real" programs from start to finish. (By real, I mean not examples, tutorials, or mini-anything, but full applications) with encryption and user details.
A decent start would be to make the classic phone-book program, writing and reading individual records from an encrypted file or database.
When you have that down and working, and preferably looked over for security by an experienced programmer, THEN you could look at networking, and then you start small and simple, make a chat program, make a mini-game like the old tank-battle games, once you have it working, add encryption, THEN you can look at writing server software.
I'm really not trying to extinguish your enthusiasm, but the reality is that a secure networked app handling sensitive data, is much beyond beginners, UNLESS they have an experienced programmer there to help them and survey their code through the whole process.