I would focus on a plan first. Threats are often benign. Don't get bogged down in what your opponent could play. Look at what you can play to achieve your plan. This might involve trades.
A simple mistake I make is to not develop my pieces. Attack less in the beginning. If your opponent is not good, you will have many chances later with more pieces, and if they are better you'll need the pieces to help defend.
I can recommend an excellent book for your purposes.
Everyone’s 2nd Chess Book by Dan Heisman
It is out of print, but I bought it off of Amazon from a third-party. It is exactly what you need.
There are also threads on this site with excellent advice.
When I solve tactics my thought process is random. This means I will jump from piece to piece until i find the corect move. Usually I look at pieces that are closer to the enemy.
I use the checks, captures, threats routine but only when I can't find the right move with my randomness. 
First of all, I think it's nice to know what really strong players are doing at the board is referencing their personal known database of patterns, ideas, and evaluations to make sense of a position and get ideas, then they use calculation to hammer out the details. Calculation vs rating is like a bell curve. Those in the middle, ~2000 calculate the most. Beginners calculate less because they're not as good at calculation. GMs calculate less because they know so many patterns and ideas they are more efficient (e.g. they can discard many bad moves on first sight).
But ok, on to your question.
There is a lot of good stuff out there, e.g. Heisman, so I'll try to go at it from a different angle.
In an equal position, where the opponent isn't making an immediate threat, essentially I think the process goes like this:
1) People look to see if they can win (checkmate or win material)
2) They look to see if they can make a useful threat (e.g. force defense that causes loss of time, passivity, pawn weakness, etc)
3) They try to improve their position (or hinder their opponent's)
The specifics of the above are what you're really asking about, but this little outline I think may be useful in that most beginners get stuck in the middle of #2. If they can't make a useful threat, they're unsure, and so they lash out and make any type of threat. If it can be ignored or defended in a way that improves the opponent's position, then it was a mistake.
Now, the (not so) funny thing is you can't really make use of the advice to keep going on to #3 until you have enough experience and knowledge to better judge which threats are good and which are a waste. But I think it plants seeds for future growth. Specifically: try to use all your pieces. In the opening we're taught to get all our knights and bishops and queen off the back rank and into the game! But this concept lasts all game long. If you're attacking on the kingside, you don't have to knock them out immediately, first you need to bring as many attackers as you can.
Similarly, as well known quote goes: "One bad piece makes the whole position bad" and another goes something like this: "Amateurs play by moving their best pieces, masters play by moving their worst"
Similarly, a useful bit of advice that's passed around goes: "when you're unsure what to do, find your worst (read: least active) piece and improve it (read: move it closer to the center or closer to the action)"
Anyway, if threats, captures, checks is new to you, then I'm guessing a lot of this is too. Recognizing threats, and evaluating their strength, is a big part of chess, and in a broad sense will be your main focus for a while, which is fine and necessary. But in the back of your mind remember chess is more than just making threats over and over, the strongest threats are ones made from a strong position, where none of your pieces have been left out.
Let me end by answering your title concisely: I look for a threat, and if I don't have one, then I improve my position. Generally threats come from tactical knowledge, improving my position comes from endgame and strategic knowledge.
As a pretty new player, Im still struggling with ultra basic stuff, like just thinking correctly during games. For example, it was relatively recently that I learned the whole Checks, Capture, Threats thing.
I was wondering, do you guys have a specific mental routine you go through before each move?