What are the best ways to support a new parent as a friend?

My friend has just had a baby and I am looking for the best way to support him and his girlfriend and the baby.
At the moment I can think of:

• Keeping in contact to see how they're getting on, but understanding if they're busy and can't talk.
• Offering help with things like shopping and chores.
• Offering to baby sit so they can relax for a little bit.
• Buying things like baby clothes and other baby things.

I can see lots of advice for new parents, but just not sure if there's anything else I can help with. Obviously don't want to be overbearing either.

Those are great things to offer. Keeping in contact by means of letting them know you're just there when they need a help.

I see that your post is 7 months old. Wonder whether you've figured out the answer yourself. Here's my two pennies worth:
If they aren't financially strong, you can buy diapers for the baby.
It's a good idea to buy food or health drinks good for feeding moms.
If you are very close to all 3 of them, you can take the baby for a night. That's if the baby is comfortable with you and the parents have faith in you. I've done it only for my sister. But remember that you may be exhausted the next day.
You can drop in at peak times, the times your friends need to multitask beyond human ability, and either take over some of the chores or be there waiting to be asked to do little things rapidly. For instance, you may be able to reach the baby before he/she pulls the rubbish out of the can while the parents can only give out a yell as they lift their heads from ironing or pouring a cup of tea.
Another splendid help is getting the baby ready when they are going out. This will allow both parents focus on getting themselves ready in peace. Moms tend to struggle to be ready on time or look tidy at all when they need to prepare their kids for the same trip.
Let them know you are ready to take the baby for doctor's consultation. Babies tend to alarm us into driving to the docs at very inconvenient times.
Plan gatherings where the baby is welcome and not the focus of all conversations. Let the baby not be a handicap of parents. Maybe plan a house party with all the jolly fellows while ensuring the baby isn't seen as out of place. Anything you can do to let the parents realise that they can still be themselves, do the things they love, that the baby is not always a reason to avoid what they would have normally done.
I arrange coffee dates with my young/new parent friends and borrow my nephew to give them company in creating nuisance at the coffee shop with baby wails. Just giving them company in anxiety and embarrassment.

Some days, late night chats can be a good idea. Let the parents decide whether they can have a long chat or a chat at all. My friends like company over a mobile instant messaging app while they are kept awake in the middle of the night by their babies.

You got it right when you said you won't take it to heart if they can't find time for you. That's really important. They still need you and love you, although they may not be able to show it to you these days. Sleep deprivation alone can drive you crazy. So do be forgiving.

When my friend's daughter was a baby I offered her and her husband to leave her with me for a day or two so they could sleep enough and to have some time with each other. They were so happy

For me, I think the best way to help a friend is that give her advice about parenting. the dos and donts especially if it is her first time as a parent. or if you have a free time visit her then make her feel that she has a friend that she can lean on or talk too