What thought process should one use to find candidate moves? Do you have a thought process for finding candidate moves?  

Once you find all possible candidate moves, how do you narrow it down to your final choice?  

Regarding the process, do you have a certain order of things you look at or is it in a random order?
Here is a basic checklist you should always consider (in order of priority):


Examine all tactics currently available to both players on the board within the next move; these take priority. Usually, each tactic has a purpose, such as check-mate (highest priority), winning material (secondary-priority) and securing a very strong position (tertiary-priority.)  


In the case of an opponent's tactics, search for "in-between" moves in the execution of the tactic. Many times one non-obvious in-between move can spell the difference between a beautiful defence and losing the game. To give an example of an "in-between" move, suppose your opponent has a discovered attack ready on your queen using a knight and a bishop. His goal is to win a rook for his knight by using the discovered attack and simultaneously attacking your rook and queen. He does not realize that your queen has an "in-between" check move allowing her to move to safety as well as your rook.
In the case of tactics available to you, also search for all "in-between" defensive moves available to your opponent. For example, suppose the previous example of the knight, bishop discovered attack on the queen was actually your tactic. After thoroughly investigating your opponent's options, you realize he can check and move his queen to safety with the in-between move. Therefore, you move your king to a square where a check is not possible, forcing the queen to move to a worse position this turn else face the wrath of the tactic the next turn.

Search all available tactics (as thoroughly as above) within 2,3,4, ... n moves where n is limited by the amount of time you have on the clock and your gut-feel for how important this position is vs. how sure you are your current best move is.
Search for guaranteed positional weaknesses in your opponent. In the lack of tactics available in the game, it can be hard to find a reason to play one move vs another. This may be a daunting task for a beginner, but rest assured there is almost always a best move. Almost all positional weaknesses deal with pawn positioning, specifically the colour your pawn chain sits on, or lack of being able to form pawn chains. In order to achieve positional weakness in your opponent, you may have to do it through a tactic so keep that in mind. I'll list some common positional weaknesses:


A knight placed where an enemy pawn can not "kick" him off is substantially stronger when placed close to an enemy king. This is called a knight on an outpost. This is mainly because it opens up the possibility of many forking tactics. Often, when your opponent has no other means of kicking your knight, such a knight is worth at least 1 rook, sometimes 1 rook and a pawn. 
Doubled-pawns create long term weaknesses since they can no longer form pawn chains in one direction. DO NOT consider this a considerable weakness in the early or middle game, because it is not, it is only a weakness in the endgame. Doubled pawns near their king, especially if he is castled, are always a weakness at every point in the game.
The colour your opponent's pawn chain sits on determines their good bishop (the opposite coloured to that colour bishop is the good bishop) seek to trade their good bishop while keeping your good bishop, this is a minor advantage that becomes greater in the latter part of the game.
The first one to double rooks on a file usually owns that file for the rest of the game. There are notable exceptions and usually a great many tactics available to both sides when fighting for dominance over a file so be careful with this one!
A rook on the same file as an enemy king or queen (even if there are many pawns and minor pieces blocking) is still a valuable position for the rook, as you can usually create interesting tactical variations by removing the pieces in the way 1 by 1.
Look for "back-rank" problems in your opponent (as well as yourself!) If your king or your opponent's king can be mated with one rook attack on the back rank this makes for interesting tactical options for both sides!
Try to make your rooks "pigs." A pig rook is a rook placed on the second to back rank (either rank 7 or 2 if you are playing white or black.) A pig rook is very strong as he is usually both attacking undefended pawns as well as posing a considerable threat to the enemy king. If you can get doubled pig rooks this usually means a win or a draw for yourself. This is generally hard to accomplish as it takes a minimum of 2 moves (usually 4+), but look for tactics as this is almost always a win especially near end-game!
Rooks belong behind passed pawns, and always push your passed pawns, the closer they get to the other side the more they are worth!
The three (or four) pawns in front of a castled king are extremely important. Enticing your opponent to move them (to kick a bishop or a knight) creates a permanent weakness in your opponent's defences. Always give your opponent the option of kicking your piece with these pawns, as when he does you know that you can take advantage of the new weakness he has created. 

With that last point, I would like to conclude this with two simple principles: the principle of two threats and the principle of every move leaves something behind. The first principle is easy to understand, create 2+ threats for your opponent and it will be hard or impossible for him to defend both. The second principle is much more subtle but just as important. Chess is a game of a delicate dynamic equilibrium. Every move you or your opponent makes gains some advantage but also creates some weakness, if ever so slight. Search for these weaknesses in yourself and in your opponent at all times.