Hi B.,
Congratulations on your upcoming baby. I'm 30 weeks pregnant, too! And I am a child custody mediator, so here is the scoop:
The father has the right to a paternity test (which I think they can do before the baby is born in some cases).
You have the right to speak with the Department of Child Support Services to open a "Paternity Case" in which you ask for help paying for the medical costs of your pregnancy and child birth, as well as help once the baby is born in the form of child support payments to cover and childcare and other expenses, as well as insurance coverage.
In the same court case you can also file paperwork to request orders for physical and legal custody and visitation. The court will send you to family court mediation where you and the father can work out your own agreement. If you do not reach agreement on the issues, you both will be sent to court to have an order put in place.
OR, you could do all this informally without any court intervention so long as you are both reasonably civil and cooperative.
In terms of the father's contact with the baby, the most developmentally appropriate approach is for him to have frequent (daily is ideal) and short (1-2 hours is ideal) visits for the first 2-6 months. The schedule should shift to less times per week for longer duration as the baby gets older. Keep in mind that if you plan to have the baby in daycare full-time, it only makes sense that this slow approach would not apply as well, since the argument can be made that dad is just as appropriate as daycare: If the baby can handle the separation from you for daycare, why not for dad?
I read through some of the other responses and want to clarify - You can start a child support case without establishing child custody or visitation. That would take a different filing. I disagree that the court always orders a 50% visitation time-share, especially when the parents do not and have never lived together. The one important factor to consider if you have no child custody/visitation order, is that there are certain things that require both parent's consent by default, such as obtaining a passport and sometimes school enrollment. If that is important to you, you may want to consider establishing custody/visitation down the road.
A good resource for all these questions is the Family Law Facilitator's office (Self Help Center) in Alameda County. Find them on the Alameda County/Court website. They can help with child support, child custody and visitation information and the filling out and filing of papers. Department of Child Support Services does child support only. Family Court Services does child custody and visitation mediation only (no finances discussed in mediation). Good luck with this.