Psychedelics have a huge range of effects, you're not going to match perfectly with another person's experience. I recommend look at this website for a while:
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psychonautwiki.org
In term of getting rid of addiction I can attest to this, for 3 hours I don't want nicotine, it's quite freeing. I imagine for people with more willpower than me they carry that feeling and never drop it.
Curing depression, from my view depression is your stuck in a rut of negative thinking, and psychedelics take you out of this mindset and throw you into something mad for a bit and afterwards you're still the same you, same problems and maybe even the same feelings but you'll have that memory of a different mindset and emotions and perspective about everything and it changes how you think because you can't deny that you thought differently and it felt very real.
This is the real therapeutic benefit about LSD and psychedelics. They allow you to see things in a new way. If you've got something very powerful like mental illness, abuse and trauma, very negative childhood experiences or lessons, guiding your decision making in life leading to ultimately an unsuccessful and unhappy life; then having something which allows you to see things differently, maybe not correctly or permanently, just differently for a while, gets people on journey's of self reflection and ultimately improvement.
We are our own worst critics, and we're the worst at it too.
In terms of my own experiences.
How I felt about my sensory input changed, it changes what foods I like, natural foods became a preferred option but manufactured food could be nice if it was done correctly or with lots of sugar/fats. Visuals things in this world that were man made that didn't reflect natural rules on how things should look (I'm not quite sure what I mean by this, but for the most part it is obvious when a person has made something and the obviousness of it pales in comparison to nature's beauty.) Things felt amazing, sounds felt like they were further away but I could hear them more clearly. Senses aligned with other senses, as in they feel the same in their own way. Not synesthesia, think more dancing how movement links to the sounds you hear.
Time became impossible to actually keep track of, your sense of it is fucked, if you take too much you'll be unable to keep track, if you've not taken enough to lose that control over thoughts then you'll be able to work out how long it has been, this will be more difficult.
Recursive thoughts, this is thoughts coming round again and again. This can be amazing if you're having a deeply personal self reflective and transformative realisation about yourself that feels euphoric every time, this can be hell if you're having a traumatic thought about something horrible in your life and you're stuck in the loop for a long time. (Some people however, report feeling better about the things causing negative emotions after)
I got spasms throughout my body, they felt brilliant.
I would and would not recommend taking them if you have a mental illness that involves hallucinations or delusions. It will worsen your condition, it may make new things triggers and increase their frequency. If you have a history of mental illness in your family and you feel you might just have it inside you too, just don't do it.
That being said, psychedelics also helped me see my biggest issue with myself and it was regarding delusional thoughts and me even knowing I had them.
It's my honest opinion that the only people who would actually genuinely benefit from taking LSD and other psychedelics are very maladjusted people. Boys who were violent from a young age, women who've been raped, people drowning their consciousness in chemicals because they can't bear their own thoughts, all these people need a moment away from normal life, to consider other things they've never considered.
Normal, regularly adjusted people who've had the correct amount of trauma do not need them, as they will have had many experiences in their lives which have forced them to reevaluate the way they think about the world in an honest way. You'll know if you're way in this world is functional or not because you'll be consistently improving on your failures, not getting worse.
You can still get positive mental changes from them if you are a well adjusted person, but do you need them, or would life have given you the same lesson in a less intense way because you're naturally receptive to new modes of thought.
Sorry for my wall of spam, I, like all the evangelical-like long haired hippies pushing psychedelics on the world can talk a lot about our experiences and you won't learn much, but I really do think they have benefits, it's not a miracle drug, but it is better for some things than any other tool we have in our arsenal right now.