Suggestions? – calendar type plugin

I have a very informal group of swimmers. There are about twenty different lengths sessions at our local pool. We will train better if (say) three of us turn up to the same session, but our availabilities are all irregular.
I want a way for individuals to come to the site, and to be presented with a list of sessions for, perhaps, two weeks ahead (rolling) where they can just click to say when they will be swimming. Others can then choose accordingly and populate one session rather than another.

I am very open to suggestions as to the easiest way of doing this, and even better to a plug in which will either start me off or get me there.