Ok - so self-soothing is a very poorly understood term. In real terms babies are actually NOT capable of self-soothing. Let me explain....
Self-soothing is a made up term (coined in the 1970's) which actually means to regulate your own emotional state. This is what we do when we're highly stressed/excited/scared etc. We experience a pounding heart, rapid breathing, and our brain is whirring - right? When we're in this state, we do several things to 'calm down' our emotional state (or self soothe) we take some deep breaths, we have a glass of water, we talk to someone, we give a bit of a lecture to ourselves! In essence, we have the emotional and psychological maturity to get from ^^^^^^^^^here^^^^^^^^^^
But here's the clincher - this skill comes from the most mature part of our brain which carries on developing until we're about 25. This explains why you've met teenagers (or remember being one!) who get themselves wound up about exams and can't calm down or control their anxious thoughts.
You can't speed this process up. Most slightly older preschoolers are able to self soothe, but babies definitely physically can't. It is normal and appropriate for them to require the loving and gentle responses of their parents or caregivers to help them into a state of sleep.
It's true though, that some children are 'easier' to get to sleep that others. This is a combination of their innate personality traits, the regularity of their habits, how 'set' their body clock is, the intensity of their emotional response, how distractible and adaptable they are, and how intelligent they are (often highly intelligent children find it harder to calm down, and also need less sleep).
Some babies, if they are put down very calmly and they are already peaceful and in a quiet state will go off to sleep. But this is not self soothing - they have not 'soothed' themselves - they've just allowed sleep to 'overtake' them.
So don't trust anyone who talks about babies self-soothing - it shows a fundamental lack of understanding about the psychological and emotional developmental states of infants and young children.
Certainly, leaving children to cry does NOT teach them to self soothe - it teaches them to give up and stop trying. But this does not fundamentally mean they have self-soothed. In fact, in a well known study, babies who had finally gone to sleep after being left to cry during 'controlled crying' had the levels of the stress hormone cortisol (which is toxic to the rapidly developing brain) measured. They found that the stress levels were the same when they were asleep as they were during full-on crying. The implication is that crying yourself to sleep and not being responded to, means you fall asleep and stay asleep in a state of high arousal and stress.
What the aim with fostering a more independent sleep habit should be, therefore, is to help babies to be as calm as possible, and introduce some very soothing and repetitive actions which help babies to become familiar with their pre-sleep ritual.
Some helpful ideas - use as many of their senses as possible - think about their 7 senses (oh, yes - 7, not 5. You heard right!)
Sight - being reassured by you, or the association of darkness, or in older children - a nightlight (not blue light which shuts down melatonin!)
Sound - gentle, repetitive sound, such as white noise from a fan or dehumidifier, or gentle music. In older children I like meditation CDs
Touch - holding, stroking
Taste - a pre-sleep drink such as milk or water. This is why if your baby is falling asleep on the bottle or breast I would leave it for now!
Smell - use a gentle smell like lavender, chamomile or another suitable oil - check the safety of the oil in the room according to the age of your child. Use a reed diffuser or even simply spray a pillow spray just before bedtime to spritz the room
Proprioception - this is related to knowing where all the parts of your body are at one time. Try an extended massage where you gently touch every part of your baby quite firmly. Patting on the bottom to help your baby go to sleep is often helpful as well. In older children a lightly weighted blanket can really help too
Vestibular - this is your sense of balance. Hold your baby very closely and firmly. Try some very slow movements like rocking. Make sure your baby or child does not have an ear infection or glue ear - this can be very uncomfortable and upsets their sense of balance.
It is normal for sleep success to go up and down. Any program of introducing several sensory sleep cues should be expected to take about 4-6 weeks for your child to get used to. Psychologists say it takes 6 weeks to make (or break) a habit, so this seems a sensible timeframe. Adding in more cues is your first step, followed by gently removing one cue at a time over several days and weeks. Start with the most annoying cue (be it breastfeeding to sleep, rocking, bottle, dummy or whatever) and stop doing it BEFORE they are asleep, but carry on doing the other cues.
Only when they can go to sleep without it do you move on and remove another one.
So, realistically, stopping an interaction between sleep and something else that you're trying to eliminate will take at least 2-3 months. But this is a long term strategy which will have lasting benefits.
Good luck! And of course, you know how to find me if you need more help than the above post!!