しかし - it is often used when you are adding something additional that is different from what was previously mentioned, but not necessarily contradictory.
確かにこの本は面白い。しかし値段が高い。
This book certainly is interesting. But it's expensive.
The fact that the price of the book is high does not contradict the fact that it is an interesting book, but since you are abruptly introducing this different evaluation, しかし is appropriate.
ところが - it is used when the following result is unexpected or is different from what anticipated earlier.
きっと優勝するだろうと思っていた。ところが十位以内にも入れなかった
I thought for sure that I'd win. However, I didn't even make it into the top 10.
The person thought one thing (that they'd win for sure), but then a different result came (they didn't even make it into the top 10), so their expectations were different from the result.
しかしながら - it has the same meaning as しかし, but it is a more formal word, used in formal speech or written Japanese.
それでも - it is used when tentatively acknowledging the previous statement, before ultimately contradicting it.
ほんとうに何度も謝った。それでも彼女は許してくれなかった
I really apologized over and over. Even so, she wouldn't forgive me.
There are many more, so if you have other questions, it would be interesting to dissect those too.
And for completeness, my assessments of these came from this thesaurus entry, not my original thoughts.
https://dictionary.goo.ne.jp/thsrs/2222/meaning/m0u/%E3%81%97%E3%81%8B%E3%81%97/
however
what is the difference between all the diffrent types of ways to say "however", orしかし、しかしながら、etc.
and when do you use each.