Tasawwuf is the science/means of attaining Ihsan(highest level of worship) and describes different ways to purify our Nafs(the self) and the soul(Ruh).Ihsan and Tazkiya are mentioned in various places in Quran and Hadith
Tazkiya is to remove/purify all the evils/sins present in us. There are many Ayats in Quran regarding the Tazkiya.
Bayt is a kind of enrollement/agreement you make with your guide/sheikh regarding leaving sins, doing some deeds(amaal) etc. On different occasions Sahaba(R.a) did bayt with our Nabi(saws).
A Sheikh is a teacher/guide for this spiritual journey.
The path/complete course which can have different milestones and stages is called Suluk.
The ultimate aim of Tasawwuf is to attain the level of Ihsan in Ibadath and to clean/remove all the sins of a person.
There are some people who have deviated from the actual teachings of Tasawwuf. But there are authentic scholars and guides and because of some people you cannot blame the complete science or community. There are deviated people in different branches/areas of religion but you cannot leave the branches and areas of religion because of the deviated people.