How does one practice suffering well? Correct me if I'm wrong, I understand a lot of buddhism is about relieving of suffering through practicing of detachment. Removing detachment is a long process. Until one removes those attachments, there will still be suffering. 

My question is: until one removes all of these attachments, one will suffer. How does one deal with these sufferings? Should one try to escape these sufferings in the moment? How does one suffer "well"? And not try to escape the sufferings?
One will not get ride of suffering by simply trying to ignore or oppress. Attachment has craving as its cause and craving has ignorance as its cause. So one needs to gain insight, how things really are. To gain that, one does well to learn about the teachings of the Lord Buddha and then look for one self, if this is right, as well as to practice the way to clean the mind, so that it is able to concentrate it so that one is able to see.

What ever one sees, as it really is, inconstant, stressful, subject to change, one gets, disenchanted. Disenchanted, he becomes dispassionate. Through dispassion, he is fully released. With full release, there is the knowledge, 'Fully released.' He discerns that 'Birth is ended, the holy life fulfilled, the task done. There is nothing further for this world.

It's not possible to escape form suffering by developing such what is called "householder-equanimity" and suffering is actually a yogis friend, which does not mean that one tutors one self, but observer it and in this way one finds a way out without simply counteract in an unwholesome way, Mr Jason.

The practice start with things many estimate as child work, now most are not even able to make this there way of live, like practice generosity (giving, letting go material thing), Sila (virtue)... the whole palette of meritorious deeds. There one is confronted with all his/here raw defilement of greed, dislike, stinginess, conceit. One has to work against the defilements all the time, while observing them well.

So please do not fall in the current promoted misinterpretations of non-reacting and simply developing a bunch of ignorance, calling it equanimity.

Last but not least, Atma likes "How does one practice suffering?" how ever, although it most important to know it well in all its aspects, its good not to forget to develop the path out of it as well.

Each noble truth has its certain task.

A good place to begin, actually has been always the first step is to follow good basic talks: 


Better to Give than to Consume
Sensitivity through Generosity


And put this things into practice. Joy and pleasure are granted, as well as much suffering at the beginning with it.

(Note: this answer has not been given with the agreement to be means of trade or the purpose of/for trade and/or keep people trapped and bound. How you handle it lies in your sphere, but does not excuse the deed here either.)