It's a very broad question, strongly dependent on one's experience. From what I consider fast and reliable, as far as small environments are considered, you may want to take Rancher for a spin.
It's super easy to start with. What's more, there's a range of services like Gitlab or DokuWiki you can start with just one click. On top of that, you can configure a load balancer, that can perform the redirections you mentioned. I think it's one of the fastest options to get a functional and scalable stack. Definitely not the most stable one, compared to enterprise-grade OpenShift, but I think it'll do just fine.
I will not go through all the setup details as I believe it's not what the question is about, but you can start with setting up Rancher 1.6 docker server going step by step through the official doc guide. It's pretty straightforward - one bash command and you are up and running.
Openshift
is a platform competing to Rancher
. To my best knowledge, it's harder to work with, especially having no experience. It's more stable, that's for sure, alas requires more effort in general.
I intentionally omitted a few options as I took an assumption OP wants it working asap while still easily being re-configurable, stable, and GUI-manageable.
-- edit a few years later --
Rancher
and Openshift
are still actively developed and attract new users. Rancher released a stable v2
since my original answer, and so I no longer recommend looking at v1.6
.