If you already have at least one other cat when you bring your new kitten home, they probably won't welcome the new arrival with open paws! Cats are very much creatures of habit and tend to hate any change to their daily routine, and, in particular, to having a kitten pouncing on them, stealing their food and trying to share their favourite bed.
Kittens are amazingly resilient to grumpy older cats, and if they have come from a breeder or from a 'multi-cat' household, they will be used to being disciplined if they put so much as a whisker out of place. Try and arrange to collect your new kitten at a time when you will be able to supervise the beginning of the integration process, and if you are at work during the week, the weekend is often a good time, unless you can arrange to take a couple of days off during the week. The vast majority of older cats will not actually do any serious harm to a kitten - it's as if they realise that the newcomer is a vulnerable baby, and whereas there will undoubtedly be lots of hissing, and maybe some growling and spitting, and even the occasional raised paw, there will not normally be any damage done. It's just the older cat's way of showing who was living there first, and establishing who is in charge.
Although it's advisable to be on hand, try not to get upset that your new kitten has not had the welcome you would have hoped for, and don't let your existing pet think that you are cross about their behaviour, as it is a perfectly normal reaction. You should also try not to intervene with the socialising process, tempting though it may be, unless you are genuinely concerned for the safety of your new kitten. Rather than giving your existing cat or cats a 'telling-off' for not being the perfect hosts, make a fuss of them and let them see they you are not trying to replace them in your affections. You might think of it in a similar way as helping a human child cope with the arrival of a new baby, as there may well be similar sorts of jealousies and feelings of insecurity on the part of the established pet. Some extra-special food, or new catnip toys will undoubtedly help the old stager to see some benefits of having a new kitten, as will extra attention, although you may need to be a little careful as the arrival of a new kitten often starts some short-term aggressive behaviour from your existing cat as it will show its displeasure towards you and to any other pets you might have.
You may feel inclined to take your new kitten to bed with you on the first night or so, or to put it in a separate room, but quite often this slows down the natural integration process as your existing cat might well feel that the new kitten is enjoying privileges that they are not allowed! And however unfriendly the feline welcoming committee has been, the new kitten will often prefer to take its chance with the 'grown-ups' as they will undoubtedly be fascinated by their new companions. If things get a bit tough, there should always be a small corner that only the new kitten can get into until you arrive to sort things out again.
Be careful that your new kitten is getting all the food they need in those early days when they are still finding their way, and make sure that you provide them with a litter tray, even if your other cats go out. Don't worry if the kitten doesn't eat much during the first twenty-four hours, but keep an eye open for the older cats trying to steal their food, and maybe feed the kitten on her own to start with. Although it may feel as if war has been declared for a day or so, hostilities will soon cease, and after two or three days you'll probably come home and find an uneasy truce with the kitten being allowed to curl up with its older companions. Usually cats know when they are beaten, and with the more sociable of the pedigree breeds, you will soon see lots of mutual grooming going on, and will once again hear the sound of contented purring as they all curl up together in front of the fire or on the sofa.
With some of the less sociable breeds, or sometimes with non-pedigree cats, you will not always get the same level of acceptance, but nevertheless, cats are very philosophical and although they may not turn into best mates, they will certainly learn to live together in reasonable harmony.
You may not find it quite so easy to integrate an adult cat into an existing feline household, as all parties will have developed their own personalities and lifestyle, and it will take usually take longer for them to learn to live together under the same roof. Certainly you should supervise the integration process more carefully where the newcomer is an adult cat, and it is probably wise to keep them apart to start with if you are not around to smooth things over. Before introducing a new adult cat, you will need to consider how your existing cat might react from the kind of temperament they have (are they laid-back when changes happen at home, or do they tend to become aggressive or stressed-out?), as some forceful adults will not accept another adult under any circumstances, and it would be very distressing for a cat to have to be re-homed again if things did not work out. However, most cats can learn to live with a companion, and even if they do not become bosom pals, they will appreciate having some feline company for the times when you are out.
People sometimes worry about introducing a cat when they already have a dog, but provided the dog isn't a breed that is known to be intolerant of cats, or has come from a rescue centre and you have been told that they do not get on with cats, they will usually live together fairly well. A cat will often get the 'upper paw' with a dog, and will soon make its feelings very clear, but again, you will often find in the end that many will curl up together for companionship.