Since you said you're new to Bitcoin, there is something you should know:
If you don't necessarily want to run a Bitcoin node (which needs to download the whole blockchain and keep it update in a daily-basis - thus wasting a lot of bandwidth and space), you can run a SPV wallet, which is an wallet software that lets you store and spend your Bitcoins imediatelly without the whole blockchain - by relying to node servers hosted by other people. The most famous one is Electrum (electrum.org).
Alright, is Electrum just as safe as Armory is, or is it safe enough against hackers etc?
I downloaded Armory and Bitcoin Core succesfully.
But to go online I have to download some extra software.
I got some problems with that. I will show you an image.
What "extra software" did you download?
You should ONLY need to download and install Bitcoin Core... and Armory. What version of Armory did you download? and where did you download it from?

You should have Armory version
0.96.5 and you should have downloaded it from
btcarmory.com.
I downloaded 0.95.1.
Alright, I succeeded to go online with Armory. So I should be able to receive and send coins now I guess. But my balance is still 0.00 unless I bought some a couple days ago.
According to blockcypher they should be sended.