I think coming from a developer background will make becoming a 'devops' actually more tricky, your question is almost 3 years old so It would be interesting to hear how you are finding the journey, I will give an answer from the point of view of the sys admin about the applications you mentioned above and hopefully it will shed some light, or give a non technical perspective that will go some way to explaining why a person (admin or dev) would begin considering exactly what you have asked e.g. from the devops perspective whats the relationship between x, y, z are these tools greater that the sum of their parts?
I actually think sys admins have the upper hand here, most of the applications you mention in your question solve admin 'problems' and in doing so provide a more abstract data center environment, and this in turn is more programmable for developers and the new 'devops' strategy (read strategy/team, devops is not a person). So what's the relationship with the apps you mention? how does this provide a holistic approach to the IT service?
OpenStack: A tool that allows you to build your own private cloud,
hence comparable to something such as AWS
That's what it is, but what does it do? - the mostly aptly named operating system was D.O.S - it operated your disk by abstracting the BIOS, OpenStack operates your data center and abstracts your infrastructure (IaaS - is Jargon for data center operating system). Now your data center has an API, a command syntax and a GUI, OpenStack can drive hypervisors, switches, routers, firewalls, storage area networks, load balancers, docker hosts etc.. Openstack uses your hardware manufactures 'plugin' or the particular function can exist solely in software as software defined something or network function virtualization. On top of this OpenStack, and all other clouds, can orchestrate their own infrastructure by reading scripts you throw at the orchestration engine or are triggered based on rules (scale up, scale down etc.). So openstack is a giant layer of abstraction, e.g. I don't care what switch I have, give me a network with this command, or, build me a complicated load balanced, HA, publicly available, auto scaling, domain name registered, storage attached thingy - with this script I found on the internet.
Docker: A "lightweight VM", based on a few Linux kernel concepts,
which can be used to run processes in isolation, e.g. in a shared web
hosting environment.
Docker is another layer of abstraction and like cloud is a disruptive technology, it's changing the industry because it solves many operational 'problems' like software dependencies, upgrades, data isolation and sheer portability. Java became popular because of it's source code portability that developers didn't have to think about, a running JVM meant that their code should run on the coffee machine so long as it supported java. Docker solves a similar problem, to run my App you need a docker host, not, you need this version of python, this kernel, this linux distro and so on, the app still has those dependencies of course, but the underlying host doesn't care and the admin doesn't care what you do inside an isolated container (to a point). Docker is changing both the development and operations paradigm, treating an entire operating system and it's services like a binary. we can get them from a repository, version them, modify them, run them with parameters etc.
Chef: A tool to setup and configure an operating system, e.g. inside a
VM.
Yes, and not as disruptive as the first two, Chef, puppet, ansible, salt, system center operations manager and a massive plethora of other applications in this space provide a way for developers and admins to model deployments, upgrades and other actions (config changes), there doesn't seem to be any standards body over looking these efforts like there is for cloud. But we're not dealing with something as definitive as Infrastructure so, it's more painful to learn these and not much is transferable from one to the other.
Vagrant: From my understanding is to automate creation and management
of VMs: Setting them up, starting and stopping them. This can be done
using a local VM or remote, e.g. on a cloud platform.
This is the odd one out in the list of apps you mention, Vagrant is a tool for developers and a toy for admins, you can quickly stand up a development environment with vagrant, e.g. I want to develop an android app, grab an IDE from vagrant, I think it will be overtaken by Docker soon.
can you give me an advice in how to start using all this (it's quite a
lot at the same time, and I don't know yet where to start)?
This is why I think admins have the upper hand, we have had to do most of this manually and know what can go wrong, puppet manifests, cloud computing and docker orchestration will come easier to us, developers will find themselves taking many tangents so my advice to any potential devops is to be an admin first.