Right - find out before you bring the cat home what sort of food and litter he's used to, and when he's fed. Try to stick to the existing routine for at least the first couple of days.
When you bring him home, pick one room to start him off in. Choose one which is reasonably cat-proof (by which I mean that there shouldn't be anywhere where they could get stuck while trying to hide). Be prepared to leave his cat carrier in that room for the first couple of nights after he gets home as it's a safe space for him to hide in if he feels the need. It's a much better option than leaving him to pick his own hiding place, somewhere dangerous or awkward.
We tend to use our sitting room and before we bring a cat home, we put out a picnic rug which covers most of the carpet. It's common for cats to crap in the carrying box during the journey so you don't want them stepping straight out onto your carpet. We put a litter tray and a bowl of food on the rug at opposite corners, and put the carrier down where he can see both as soon as he steps out. It's important that the food and litter aren't close together - they won't eat where they poo.
We pop the carrier down, close the doors to the room, open the carrier door and then retreat to the sofa. The cat normally ventures out straight away and we sit quietly on the sofa and let him explore while we're in the same room. He'll be calmer if you leave him to sniff around and get his bearings at this point instead of trying to stroke or handle him. Let him come to you. Be in the room with him as much as you can, but usually better to be doing something other than just watching - if you're reading a book or otherwise quietly occupied and keeping an eye on him from the sofa, they are likely to be brave enough to explore.
You'll know when he's ready for a bigger space than the single room because he'll show lots of interest when you're going out of the room. With most cats, this happens within an hour or two. However, introducing him to the whole house at once would be overwhelming and while he's new, I want him to be within dashing distance of the litter tray at all times.
I wouldn't bother buying a cat bed - a nice fleecy blanket is more likely to be used and easier to wash. Do get a scratching post and if you can, I recommend the Feliway diffusers - we usually have one running when we bring a new cat home.
When he's ready to make friends, let him sniff your outstretched hand before you stroke him. He relies on smell and will trust you more quickly this way. Blinking at a cat slowly is cat-speak for smiling at them. If he's shy oor uncertain, I often spend half an hour lying on the floor (to be smaller than the cat), which is ridiculous but effective.
I have other tips for violent/feral/hermit cats, but a good shelter won't let you adopt one of those without support.