All ruminants should be given something stalky to stimulate rumen development within the first 7 days. They'll only pick on at it at first, of course. Some folk prefer to use a good barley straw, rather than hay, as there's less chance of it scratching the internal membranes. We bed our calves on barley straw, so they always have that to lick at, and give them a little soft, fresh hay at least twice a day - replacing what was there, which can become bedding or be given to older stock.
Most of ours start to show interest in cake at about 10 days old but don't really get stuck into it until about a month old. We use a calf starter mix (and have spent some time and money finding one they really like; for us it's the Champion one, companion product to Champion Tup & Lamb.) Again, just offer a little, and remove at the next feed, replacing with fresh.
As to being outside, young calves aren't good at withstanding rain, so when they can go out depends partly on what shelter is available. If none, then the weather needs to be warm and dry, really. And I'd be bringing orphan calves indoors overnight and if the weather is wet, because they haven't got mummies to make sure they're somewhere sheltered. Always well-ventilated, though, for cattle.
Introducing grass may need to be done slowly, so that the rumen adjusts to the new feedstuff. We have a small rather bare paddock that gets used for lambs and calves to start learning about eating grass.