First, Buddha taught a path for us to follow based on things we can see directly and work with and test ourselves, and to keep an open mind about the things we can’t see or know ourselves. So we don’t have to believe in rebirth, it’s not like a creed. Indeed best to keep an open mind if you can, even if you have that belief in rebirth, as many Buddhists do, to recognize if you don’t actually know it for sure, that you don’t really know what happens when you die.

This thing about having an open mind includes testing our own beliefs too. So you don’t just go by your own preconceived ideas either, because they seem familiar or persuasive or feel right. That’s where many Westerners particularly go wrong, the idea “this feels right so this is the Buddhist path” - Buddha warned about that in his Kalama sutra.

It’s about continually recognizing what you don’t know and opening out to new possibilities. So particularly also about death, to recognize it too, if we don’t know for sure what happens when you die.

As for the actual process of going from one life to another, different schools of Buddhism say different things. Not as something you have to accept as a creed, but something many Buddhists believe to be true.

So Therevadans think that the next moment after your death in this life is the first thought moment of your next rebirth. Tibetans think that though this can happen, there often is a long period of several weeks in the “Bardo” state - a state of transition, confusion, bright lights and loud sounds before you end up in the next life, elaborated in detail in the Tibetan Book of the Dead.

We don’t feel there is any need for a soul or atman to do this, any more than there is a need for such for a child to grow up to be an adult.

I’m not sure where this idea that Buddhists think everything is illusion comes from. Our path is to do with connecting with whatever is true and wholesome. We can only do that if there is some truth to be found or realized or connected to. However along the path we may find that we have fooled ourselves in many ways. Especially, we tend to treat fluid changing things, as if they were fixed and permanent, not letting them change, or not letting ourselves see that they change. So, it’s the idea that there is some truth to be realized there, by recognizing this, this fluid changing nature of things. So that’s the illusory nature, or Maya on the Buddhist path, becoming aware of this impermanence of things and not running away from it but letting ourselves see it, really see it, letting it really soak in.

Also, Buddhism isn’t really about trying to get a good life through karmic cause and effect. It’s not against that either. Buddha encouraged us to be kind to ourselves :). If we can find a way to be happy in this world for a while, there’s nothing wrong with that so long as it’s not harming others. It’s good. Good to be happy, good to help others to be happy.

However, for us, karma is not something imposed by any external being. It’s just cause and effect. Prayudh Payutto, uses the example of climbing a flight of stairs. So there are many consequences of going up a flight of stairs, you can’t touch the ground, you may be a bit out of breath, you may have a better view etc. All that is karma.

This is what he wrote:

There are three philosophies which are considered by Buddhism to be wrong view and which must be carefully distinguished from the teaching of kamma:

1. Pubbekatahetuvada: The belief that all happiness and suffering arise from previous kamma (Past-action determinism).
2. Issaranimmanahetuvada: The belief that all happiness and suffering are caused by the directives of a Supreme Being (Theistic determinism).
3. Ahetu-apaccayavada: The belief that all happiness and suffering are random, having no cause (Indeterminism or Accidentalism)
   
   See Misunderstandings of the Law of Kamma

There one of the biggest western urban myths about karma in Buddhism is that first one: “The belief that all happiness and suffering arise from previous kamma (Past-action determinism).” which as he says is a wrong view in Buddhism. For more on this my Karma in the Buddha's teachings

So, karma is like this ordinary cause and effect we can all see, when we walk up stairs, or go to the shops or put on a kettle for a cup of tea etc. Buddhists just take a rather broader view on it than most of us, including this idea that there could be cause and effect like that operating from previous lives and through to future lives. Like, to take Prayudh’s analogy - that you climb a stair in a previous life and arrive out of breath in this :). I mean not literally, but sort of like that, that there can be effects from previous lives or whatever it was came before you were born that continue into this life. But in a rather ordinary way in a way. It’s not really the idea that every single thing that happens to you in this life, you can trace back as a response to some particular past event in a previous life. Any more than you can in this life. As Tai Situ Rinpoche put it once, it doesn’t mean that if you are bitten by a mosquito on your nose in this life, that you must have bitten that mosquito on its nose in a previous life :).

Some of the Buddhist schools pay more attention to that broader view of karma, while others, such as Zen Buddhists, hardly pay much attention to it at all. But even with the Buddhist schools and teachers that put a fair bit of emphasis on teachings about karma in this broader sense, it’s not really the central teaching of Buddhism.

The central teaching is much more to do with opening to others and - basically not having such a closed in claustrophobic approach to everything.

Indeed Buddha taught that you can never escape from the closed in situation we are in by this process of finding causal conditions to create nice peaceful and happy conditions for yourself. Anything that is conditioned like that is also something that can cease when the conditions for it go away. So - basically his central teaching is that the process of working with karma like that can never free you from this claustrophobic world we get caught up in, this wheel of Samsara.

As Walpola Rahula put it, in his “What the Buddha Taught”:

"The Buddha does not deny happiness in life when he says there is suffering. On the contrary he admits different forms of happiness, both material and spiritual, for laymen as well as for monks. In the Anguttara-nikaya, one of the five original Collections in Pali containing the Buddha's discourses, there is a list of happinesses (sukhdni), such as the happiness of family life and the happiness of the life of a recluse, the happiness of sense pleasures and the happiness of renunciation, the happiness of attachment and the happiness of detachment, physical happiness and mental happiness etc.

“But all these are included in dukkha. Even the very pure spiritual states of dhyana (recueillement or trance) attained by the practice of higher meditation, free from even a shadow of suffering in the accepted sense of the word, states which may be described as unmixed happiness, as well as the state of dhjana which is free from sensations both pleasant (sukha) and unpleasant' (dukkha) and is only pure equanimity and awareness—even these very high spiritual states are included in dukkha. In one of the suttas of the Majjhima-nikdya, (again one of the five original Collections), after praising the spiritual happiness of these dhyanas, the Buddha says that they are 'impermanent, dukkha, and subject to change' (anicca dukkha viparinamadbamma). Notice that the word dukkha is explicitly used. It is dukkha, not because there is 'suffering' in the ordinary sense of the word, but because 'whatever is impermanent is dukkha' (yad aniccam tam dukkham). "

So - working with karma does not give us a “way out”, not for Buddhists. It helps us find stability in peace for a while in this world. It’s important. To recognize that actions have consequences. To learn to live in this world in a way that first doesn’t harm others, and if possible helps them too.

But opening out to others and compassion and loving kindness - that’s part of the path and is a way that we can transcend all that. That’s in all the main schools of Buddhism. Along with humour and not taking ourselves too seriously, and the help of friends to bring a perspective we can’t see easily for ourselves.

It’s a path of connecting to whatever is authentic and true. Down to Earth and straightforward, and the aim is not to enter into a mystical state or a trance or anything like that.

As Walpola Rahula put it:

"It is incorrect to think that Nirvana is the natural result of the extinction of craving. Nirvana is not the result of anything. If it would be a result, then it would be an effect produced by a cause. It would be sankhata ‘produced’ and ‘conditioned’. Nirvana is neither cause nor effect. It is beyond cause and effect. Truth is not a result nor an effect. It is not produced like a mystic, spiritual, mental state, such as dhyana or samadhi. TRUTH IS. NIRVANA IS. The only thing you can do is to see it, to realize it. There is a path leading to the realization of Nirvana. But Nirvana is not the result of this path.You may get to the mountain along a path, but the mountain is not the result, not an effect of the path. You may see a light, but the light is not the result of your eyesight."

In all the main schools of Buddhism again it’s a path of balancing wisdom and compassion. There wisdom is not intellectual wisdom but rather, seeing the truth of things, seeing clearly, having a good clear understanding of what needs to be known. If you focus over much on wisdom you can be a hard hearted person, who sees things clearly but without much feeling or connection. While if you focus overmuch on compassion you can be a good hearted fool. So it’s a case of balancing both. Tibetans have the analogy that it’s like a bird’s wings, a bird needs two wings to fly, so as a follower of the Buddha’s path you are flying on the two wings of wisdom and compassion. So I’m not sure if that analogy of a bird is in Therevada too, but the idea of balancing wisdom and compassion is.

See also my answer to What is the difference between Buddhist and Christian compassion? - I’ve used part of it in this answer.

Also I think most Buddhists wouldn’t say of themselves that they are atheist, or theist for that matter, or agnostic. It’s more that if you are following the path of the Buddha, there’s no need to speculate in any direction about such things. Trungpa Rinpoche called it “non theist”. See his How To Become a Buddhist by Taking Refuge by Chögyam Trungpa Rinpoche