The most effective time to train a cat is when they are a kitten. Kittens are much more malleable and formative, and can be shaped into the type of cat you want as an adult. There are many different things that you can do with your kitten to train them. If you want your adult cat to be cooperative for being picked up and carried around, then do this a lot with your kitten. Periodically, pick your kitten up and carry them around the house while snuggling with them and cooing at them. This is also a time when you can give them treats to reward them for being good about being picked up. If they struggle and try to get out of your arms, try to distract them with food or a toy while you are holding them. Always reward in a positive manner, never ever punish them for not doing what you want them to do.
If you want your cat to be a lap cat, then when your kitten is worn out and ready to crash, or when they are already sleeping, pick them up and put them in your lap. Don't force them to stay, but if you do this often enough, eventually they will start seeking out your lap when they are tired. Keep in mind, once they are on your lap, you will need to stay there for a while. It isn't fair to your kitten to get used to snuggling on your lap, only for you to get up 5 minutes later consistently. If you do that enough, they will stop seeking out your lap.
If you do not plan to declaw your cat, but prefer to keep your furniture intact, you need to start working with your kitten on using a scratching post as soon as you get them. Have more than one type of scratching post available so that your cat has options to choose from. Some of the more common types include carpet, sisal, cardboard, rope, wood, etc., and in both horizontal and vertical forms. Put catnip on the scratching posts to make using them a rewarding experience for your kitten. If you catch your kitten using one of the scratching posts, praise them and give them a treat. To keep your kitten from using the furniture, put deterrents on it such as double-sided sticky tape anywhere your kitten can reach. Most kittens do not like to stick to things, and will find something better, such as the scratching post. Stinky sweet smells like vanilla and peppermint may also help to deter your kitten.
You can also scare your kitten if you catch them scratching on furniture, but be careful to try and keep them from realizing that you have anything to do with it, so they don't fear you. You also have to catch them in the act, even a few seconds later, and they will have no idea why they are in trouble. Most kittens will stop what they are doing if they hear a loud, scary noise. One way to do this is to have a glass jar with change or nuts and bolts in it, and shake it if you catch your kitten scratching in undesirable places. Squirt guns and spray bottles can also be used to get your kitten we if they are scratching somewhere they shouldn't be. Again, though, make sure you only do it while they are in the act, and one squirt is usually sufficient. The goal is just to make it an unpleasant experience, so they won't repeat it. Don't soak your kitten, they may start to like it and think it is a game.
Plants are not a good thing to have around kittens. The dirt is just too tempting as a play box, and the plant itself may taste good to your kitten. A LOT of plants are toxic to kittens, so if you cannot find a different home for your plant, it is best just to make it a hanging plant and pick up any leaves off of the floor immediately, as even these can still be toxic to your kitten. You can find a list of plants that are toxic to cats at http://www.aspca.org/pet-care/poison-control/plants. If you just absolutely cannot remove your plant from where your kitten has access to it, and you have made sure that it is not toxic to your kitten, one way to keep your kitten from digging all the dirt out of the planter is to put large pebbles over the dirt that are too big for your kitten to dig out. You can still pour water over these to water your plant, but your kitten cannot access the dirt underneath. Sometimes there is no training that can be done on the kitten’s part, instead, it is on your part, such as being trained to keep certain items away from your kitten.
Most kittens will tend to instinctively use a litter box, especially if you have another cat or kitten in the home that is already using it. But if this is your first kitten and first litter box, then you may need to work with them just a little to get them to use it consistently. Your kitten should not have to travel more than a few rooms to find a litter box, even if this means that you may need multiple litter boxes in your home. You can always gradually move them and combine them as your kitten ages. You should also have the litter boxes in areas that are low traffic and that do not have any scary noises next to them such as a washer/dryer or HVAC unit.
If your kitten is having problems using the box, then keep them confined in one room with a litter box in one corner, food and water in another corner, and a bed in another corner. Don't clean the litter box immediately; let your kitten get used to the odor of stool and urine in the litter box to be able to associate it with eliminating. Don't go overboard on that though, if it is too full of stool and urine, your kitten won't want to use it. If your kitten goes somewhere other than the litter box, place the stool in the litter box and clean the area thoroughly with a cleaning agent designed to eliminate urine smells. If you catch your kitten using the litter box, once they are done, praise them and give them a treat. Don't interrupt them in the litter box, that could make them stop using it. Once your kitten is using the litter box consistently, you can expand their living area.
If your kitten is a biter, then they were not around siblings long enough to learn good manners. If your kitten is still young, socializing them with other kittens or getting another kitten may help to curb this behavior. If your kitten is getting older, then you may need to do the work yourself. If your kitten is biting so hard they are drawing blood, then this needs to be curbed much more quickly. If your kitten is just play biting and not breaking the skin, this is still something you want to discourage, but you have more time to work with them. If your kitten is biting in play, then play is over as soon as they start biting. Don't yell at them or punish them, just walk away. They will eventually figure out that biting leads to lack of fun. Don't play with them with your hands in such as way that they think biting your fingers is part of play. Just like with scratching post-training, a loud scary sound or a quick squirt of water can be enough to let your cat know that the biting was not a good thing. This may require another person to help so that the kitten does not associate the negative reinforcement with you. If your kitten is behaving well, and is not biting at a time when they might have previously bitten, reward them with a treat and praise.