To Some Extent, Chess IS Your Thought Process
"How do you improve your analytical and evaluation skills?"
This is a big question, in my mind almost equivalent to "How do you improve at chess?" since the single biggest factor correlating to your playing strength is your analytical ability. Moreover, chess is a thinking game, so improving how you think must be very closely related to improving how you play. Sure, things like having a better attitude, overcoming fear, building stamina, and other non-thinking attributes also affect your strength, but thinking is obviously at the heart of the matter.
One of the tenets of Q&A with Coach Heisman is to refrain from asking questions which entire books (or at least chapters of books) are written to address. "How to improve analysis and evaluation skills?" is one such question.
On the show I mentioned GM Andrew Soltis' work How to Choose a Chess Move, which also contains reference to the classic older work Think Like a Grandmaster by GM Alexander Kotov. So answering such a wide question on the show in a minute or two can hardly do justice to the subject.
It is difficult to discuss this subject without defining the terms, especially since in this area different authors often mean somewhat different things when using these terms. So here are my definitions:
Thought Process - the sequence of events that occur when you think about a chess move (including what happens on the opponent's thinking time). This is NOT the same as your thought content, which is what you are thinking. For example, a process could start with asking yourself "What are all the things my opponent is trying to do?" and "Is my opponent's move safe?" while the content is the things the opponent is trying to do and the information about the safety of his move.
Analysis - the process of creating "move trees" during the thought process. This can be very specific move trees, such as "If I play Nxf7, he can re-capture ...Kxf7 or Bxf7" or more generalized generation of move candidates, such as "My opponent has played 1.a4. Which openings can I play in reverse such that 1.a4 is not helpful, or even a detriment?"
Calculation - The analysis of forcing moves sequences, almost always involving captures and recaptures, giving and getting out of check, and making and meeting threats. That's why I tell students that the forcing moves are checks, captures, and threats. Much of analysis is not forcing sequences but, when that occurs, the analysis becomes calculation.
Evaluation - Determining which side stands better, how much, and why. This almost always occurs at quiescent nodes in the analysis tree (e.g., you would not stop after capturing your opponent's queen and say you were ahead a queen if the opponent can simply recapture to make the position even and "quiet"). The exception is that sometimes quiescence cannot be reached, usually in speculative sacrificial lines, when you have to use your judgement to evaluate at some point without quiescence.
There is so much written about improving your analysis skills that books are sometimes written about individual aspects. For example, even simple tactical puzzle books contain common safety patterns that help you analyze the safety of actual and potential moves.
Or, if you want to know exactly what went on in a player's mind when he was thinking, you might want a book that contains such protocols (written records of "think out loud" exercises). The three books that contains such entire protocols are de Groot's Thought and Choice in Chess, Aagaard's Inside the Chess Mind, and my The Improving Chess Thinker. A book like The Amateur's Mind does not attempt to show entire protocols, but it does contain snippets as the students answer specific questions about positions given to them by author IM Jeremy Silman. So that's still very useful even if it's different.
With so much material on improving analysis, maybe I can just say a few words about improving evaluation.
In the "old days" it was said that computer chess engines analyzed well but were poor evaluators of positions. No more. With some exceptions, engines evaluate far better than I do, even in positional positions. So this leads to an obvious exercise to improve evaluation:
Select fairly "quiet" positions where there are very likely no tactics. You can do this randomly from game books or databases, or any means you wish. Then evaluate the position, trying to clearly state which side you think stands better, how much (e.g. equal, white is a little better, white is a lot better, white is winning, white is winning easily, etc), and why. Then give the position to the engine and let it analyze and evaluate for a while (a minute should be enough) and compare your evaluation to the engine's.
For example, you might have said that White is much better because he is ahead a pawn but Black has some compensation. If the engine's evaluation comes in between 0.5 and 1.0, then you were likely correct. But if it's close to zero, then that means Black has full compensation for the pawn and the engine thinks the chances are about equal. If the engine's number is above 1.0, then it means Black has "negative" compensation for the pawn, meaning White has the pawn and a net of other advantage, too.
While the engine can't give you a "why", once you know its evaluation you can likely re-think your evaluation if it is off: why did I think that one side or the other was better than it is?
Of course, if you do accidentally pick a position with a tactic and you don't see it, the evaluation will be dynamic and depend on that tactic, so in that case your evaluation might be way off. But that would not be an evaluation error, in a sense, but rather just a lack of tactical vision (and believe me, an engine's tactical vision can be quite incredible, so don't feel too bad).
This is not the only evaluation exercise, but it's a pretty good one with almost infinite possibilities and it's fairly inexpensive, given the proliferation of free engines.
Q&A with Coach Heisman is on Chess.com TV the first Friday of each month from 5-6:30 PM ET.