What to do when you are invited to another university?

Offering to give a presentation about your work sounds like a very good idea. I would suggest contacting the professor as soon as possible saying that you would be keen to present your work. By giving them early notice, you give them the chance to either arrange something for you, or to pass your details onto anybody in the wider department who may run the sorts of seminars/colloquiums where such talks often take place.

In terms of other things you could do, if they've already told you that you'll be introduced to other members of the research team then that sounds like a lot of that will be organised for you. Another thing you may want to consider is to see what other regularly seminars, journal clubs, research groups, etc. already exist that might be of interest to you. If this is somewhere that you may have the opportunity to go and work at then you want to have the fullest sense of what your working environment would be like.

As for putting it on your CV, I'm not entirely sure that a visit is entirely relevant. If it was a visiting job then include it, but something casual like this probably not. It is the sort of thing that may be well worth discussing (if it's relevant) in a cover letter or supporting statement that goes alongside applications.

My CV has a section for talks that I've given at various institutions. You can definitely mention this under such a category.