What's the best way to study tactics

Solve the puzzles from the book diagram or set the position on a real board? And, if I already solved the puzzle after seeing it from the diagram, should I set the position on a board too, to learn the pattern?

Real board or display doesn't matter much if you ask me.  I think the key here for me at least is:

1)exposure to the concept

2) understand the concept

3) practice recognizing the concept (various problems all same concept)

4) incorporation into overall conceptual repertoire (problems with this and other concepts already learned).

I think it is important to be able to separate tactics into themes for this reason.  So, say you have a subscription at a site that allows that.  You could pick a concept, make a problem set for that concept, and do a problem.  After you do it, make sure you understand how it works, fully.  Then continue the problem set until you are very comfortable recognizing the theme when you know what to look for.

Finally, add this theme to a bigger working set of problems full of concepts you've already done this with.

Solve problems tactics problems mates combinations directly from books without moving pieces.

Get a book like Alexander Alekhine's Best Games,Nezhmetdinov's Best Games,  The Life and Games of Mikhail Tal, 1000 Best Short Games of Chess, Shirov's Fire on Board, Lenoid Stein Master of Attack, The Chess Games of Adolph Anderssen also recommended is Charousek's Games of Chess by Phillip Sergeant every move this master makes is designed to attack and take a big bite out his opponents ass he is a real doberman pinscher and does not let go when he has a firm grip:

http://www.amazon.com/Charouseks-Games-Chess-Biographical-Introduction/dp/0486258327/ref=sr_1_1?s=books&ie=UTF8&qid=1386783766&sr=1-1&keywords=charousek

Get a chess board and clock and go over all the games in the book 1 by 1 fast every day speed train!

Most tactics are best solved without needing a board and pieces, even in a book.  However, some of the more advanced instructive books like Magic of Tactics (I think that's the name) and Le Moir's How to Become a Deadly Chess Taction you'll want to play out the games but do the problems in your head, only using a board and pieces to go through the answers after you've committed to a primary variation.  A primary variation what you think will actually appear and think is objectively best so if your primary variation is eight moves long then those are what you think are the eight best moves for both sides. 

Actually, get Aagard and Lund's Right Decision CD, they have plenty of calculation exercises and even positions for you to play against the computer.  While most positions are attacking or winning you have some defensive tasks in there too. 

So what should I do?

Doing anything is better than doing nothing.

Get a free lesson from my game collections at :
http://www.chess.com/forum/view/chess-openings/theres-no-draw-by-repetition?page=1