Apart from the practical management of baby itself:
1. The number one mission of dad is to support mom in every possible way, emotionally, physically, logistically, spiritually. Expect nothing in return. No comfort, no consolation, no recognition, no affirmation, no nookie. And little sleep. This is the time when we men are challenged to become real men and learn to put ourselves and our own needs at the end of the list (or even better, scrap the list). Be a rock of stability, good cheer, service, tenderness, gentleness and strength. This will be one of the hardest things you will ever be called on to do, and for everyone involved very much depends on how you rise to the challenge.
2. The greatest gift you can give your child is for you to love its mother in utter fidelity and commitment. It is your union that is the origin of the child, and if that union is threatened or compromised, a deep existential angst is awakened, which will wound the child for life ... for the child will never truly in its deepest heart believe that you really love her/him if you reject/fail its mother.
3. Enjoy your child. And show it constantly. Hold, love, cherish, play, touch, laugh, laugh, laugh. Even when you're dead tired from lack of sleep and emotionally exhausted from being nothing but a chequebook for others.
Fully Agree with 1. Number 2 though, the words sound poetic, but eloquence does not make for truth.
You make it sound as though not loving the mother will make the child not believe that you love it. A person, young or old. Gauges how much you love them based on the love you show them, not the love that you show to someone else that they love.
The greatest gift you can ever give your child is to love IT completely. The second greatest is to ensure that it grows up in a happy home. Obviously committing your every effort to love the mother of your child is necessary, but not specifically for the child's benefit. That is more for the benefit of your relationship, as we can assume that most people are having kids with their chosen "life partner".
Now to add the flip side.
1. Mom, while on maternity leave, needs to understand that dad is under immense pressure as well. As hard as a baby is to take care of (trust me I work at home and actively see and help during my working day, so I really do know, I am not just running my mouth), mom can get into some kind of rhythm because it's all she knows all day long (Which has its own type of stress I know).
Dad has work stress, money stress, traffic stress and baby stress to deal with. It is harder to get into a routine with that many different factors involved. That doesn't mean he gets to come home and put his feet up, but perhaps a 10 or 15 minute chill out from the drive home before expecting him to dive right in. Let him adjust quick.
2. Sex. Like it or not, we are still sexual beings. As much as it is good to say that men must not expect any nookie. And they really mustn't. The relationship needs intimacy of some kind. The days roll into each other so quickly that you can easily hit the year mark before you realise that the last time you were intimate with each other was possibly some point during the pregnancy if that. You run the risk of becoming room mates that look after a baby, instead of a couple. Sure love transcends the physical and all that mumbo jumbo. But just because it can, does not mean you should take it for granted. It is far too easy to get comfortable with the idea that a baby is such hard work that you cannot even spare time for intimacy.
Make the time, once a week, once a fortnight, whatever works for you. But purposely make the time to get intimate. If, as a woman, you still hurt from either an Episiotomy or a C section. Then explain where the boundaries are. Even if it is a little "foreplay" in bed, it will help your relationship in the long run.
Obviously I don't mean the first week you get home, or even the first month. But don't let the lack of intimacy become the status quo.
3. To solidify Arthur's point here. Laughter really works with a child of any age. It wont always stop them from crying. But even if laughing out of frustration, it will still have the same effect on you. I cannot tell you how many times in the first few weeks. I was found laughing with a crying baby in my arms, at 2am. It helps to maintain sanity, and when they get older, they simply love the sound of laughter.