On the first path you are likely going to be just fine with VirtualBox, VMWare Workstation or Hyper-V (Windows only) / Parallels (Mac intended). Which one you should pick depends on your desired use of the machines.
On the second path you would go with a solution that deals with the nitty-gritty details, such as Proxmox, oVirt, Hyper-V, ESXi, or any of the other many available options - granted you are not going full cloud-based, which opens up a whole lot of different options too.
You would generally never need to worry about which components are needed where and why. I've had to worry about it once or twice before, because I've had to debug why an oVirt solution was not behaving like I wanted it to behave. Knowing the inner workings helps in that case.
As a Linux user, why would you want to use VirtualBox or VMWare Workstation? They are not so well integrated with the system, and, frankly, VirtualBox is more of a toy VM player... just go for virt-manager. It gives a conceptually similar interface to VirtualBox, but better integration with the rest of the system. Especially, when it comes to stuff like sending different key combinations.
I honestly cannot think of a single benefit to using VirtualBox (and I'm less familiar with VMWare player) compared to virt-manager. My guess is that it's more often used because it's also a common choice on MS Windows, so, you get more hits if you are going to search the Web for questions associated to VMs / you'd get tutorials for how to set up a VM that use VirtualBox. But, if you apply yourself to learning how either one of these works, you'd see no reason to choose it.