What should I do first, arrange a visit with him or see the lawyer?
Kids first. That's your new mantra.
How soon are cases like this decided?
It's not unusual for a case to take 18 months or more to get in front of a judge for a final hearing. However, what you need to realise is this 'case' is never decided permanently and will be ongoing until the child reaches emancipation. That doesn't mean you will necessarily be in court the entire time but it does mean that once begun the court will retain jurisdiction the entire time. Custody cases are never decided with finality. This is actually a good thing because that means there is always a chance for stubborn dads that will not be shut out of their kids lives.
How can I overcome the mommy and baby bias?
Not unlike how many others overcome biases of all types. Do your thing and do it well. That doesn't mean the bias will go away and I think you can make some pretty direct comparisons between the discrimination suffered by blacks and gays and those of dads. However, the discrimination suffered by dads is more ingrained and institutionalized.
How can I get overnights as soon as possible, so that I can show the courts I can care for him from morning til night?
The sooner you begin the better. Personally, I think you are better off seeing your child at every opportunity available while you build your track record of involvement before you start filing for hearings and sending legal scuds that are sure to create conflict. It's is obvious from your own retelling of the events that the mother is willing to work with you and is looking to you for involvement. That is a good thing, regardless of how you feel about her.
Should I start contributing money for medical or diapers? Should I just buy the stuff or give her the money?
By all means, buy your baby stuff. Be involved by more than just cash. Ask mom what kind of diapers she's been using then go buy a bunch. Buy rattles, squeeky toys, cute little outfits that day "Daddy's boy" on the front, a little rocker swing, pacifiers and whatever else you think your baby boy needs. That's what dads do.
Things were looking pretty bleak for me and I know I seemed like a stubburn a-hole to many if not most of you. Geting excited now that I feel like might still have a little control over my future.