Should I still talk about more salary when I think it is already good.
I need more vacation time.
You should negotiate for the best deal you can get, keeping in mind that if you push hard they can drop the offer.
If you "need" more vacation time, you can ask how flexible they are on vacation. In some companies and some positions, they are willing to negotiate more weeks of vacation. In others they are not.
If you "need" more vacation time, but don't need more salary, then my suggestion would be to negotiate solely for more vacation. When I make an offer to a candidate, I'm more likely to go the extra mile for one item, than for a raft of items (assuming that the items are under my control and that the candidate is worth the extra effort).
Have a discussion with the hiring manager that starts "I'm really excited about working here! I like pretty much everything about your offer, except the vacation time..." and see where it goes from there. That's where your negotiation skills come into play.
Seeing a laundry list of counter-offers might make the hiring manager feel as if you are just playing games or are high-maintenance.
Also, one really weird thing is that the medical benefits will start
after 3 months of service. I have never experienced that in any of my
jobs. What is a person suppose to do for 3 months who has a family?
That is rather unusual in my experience.
You could ask the hiring manager (or whoever made the offer) what new hires usually do in this circumstance.
I would assume you would need to find and pay for your own private insurance for the 3 months, or use your spouse's benefits.