Pakistan Defence

That is complement bro!.
There are no books on Sharia law which you could directly understand, that is not a comment on your intellect, that is because to understand Sharia laws you need to understand a lot of history of Islam and Prophet S.A.W's time. Thats why

only the people of prior knowledge can understand it.
But anyway, I do not know of those books in english but I can provide you the ones in urdu, "Behashti Zevar Masail".
One book that I know of in English is "Enjoy your life" by Al-Samd Al Arfi, it is about Seerat-un-Nabawi S.A.W -- life of Prophet S.A.W in other words.

So if wanted to know the punishments for different sins/crimes, like murder, rape or corruption in respect to Sharia Law, would there be a book that can help me

Bro,
There isnt a single book, there are many. Islamic laws are vast and cover smallest of details, that is the reason that you have to study 8 hours a day for 8 years to be an alim and that doesn't grant you any authority either, you have to then

study more and become a mufti!
Anyway, to understand these mentioned laws you just need to study Quran, these are fortunately mentioned directly.
However, "corruption" and "sins/crimes" are a very vast terms, you will have to be more specific.

The reason i ask is that one of my family members told me that according to Sharia law, the punishment for murder, rape and even corruption is death. I just want to confirm if that is correct.

No,
Punishment of Murders are two:
1.) Death.
2.) Diyat -- The family of the victim choose to forgive him in exchange of a financial compensation... Raymond davis was freed on this law. And this law is abused a lot in Pakistan.
Punishment of rape is "whip".

Punishment of corruption depends on what corruption he did, if the person is authority/politician/leader and stole public money then it falls under the law of "theft". The person will be sacked from his authority and for theft his hands will be amputated and will not get to Jannah (until forgiven by ALLAH).
If he killed someone, same rules apply except instead of theft, it will be the case of "murder".
And so on

Understand Islam and life of Rasool Allah (saw) and you will understand sharia law. You have to research on it, there's no any book on it. I think.
And there are countless books on life of Rasool Allah (saw)

Ps: sorry to intervene.

Also everything is discussed in Quran. So, it's best to start finding the particular ayah' of Quran related to the subject and then research on it........reading different Tafsir.

You are right about that sister, but I would add, dont study Quran/Islam on your own, it turns you into the likes of a few people that hate a particular sect without even understanding what that sect is