Every board here can be cracked with logic alone, with no trial and error. The whole game revolves around two numbers for each row, column or region: the remaining target (the printed target minus what you have already kept) and the undecided cells that are still open. Here are the tricks, roughly in the order you will reach for them.
Finish the obvious lines first
Scan for lines that are almost decided. If the undecided cells of a row add up to exactly its remaining target, all of them must stay: there is nothing left to throw away. And once a line has reached its target, everything still open in it has to go (the game even crosses those out for you). These two observations alone usually open up the board.
Cross out what cannot fit
A cell bigger than the remaining target of its row or column can never be part of the answer. cross it out. This is the quickest way to make progress at the start: look for large numbers sitting in lines with small targets.
Spot the indispensable cells
Now flip it around: imagine removing a cell. If the rest of the line could no longer reach the target, that cell is indispensable, so keep it. A line whose numbers only just cover the target is full of cells like this.
Do the remainder math
For a stubborn cell, ask two questions: can the target still be reached with it, and can it be reached without it? List the sums you can build from the other open cells: if only one of the two answers is "yes", the cell is decided. With three or four open cells this is quick mental arithmetic, and it is the single most powerful habit in the game.
Watch for odd ones out
Sometimes all the open values in a line except one are even, or all are multiples of three. Then the remaining target tells you exactly what the odd one must do: an odd target built from even numbers needs that one odd cell; a target that is already a clean multiple may rule it out.
Use the shape (one-island boards)
On boards where the kept cells must form a single connected island, geometry becomes a clue of its own. A cell that gets walled off from the kept area by crossed-out cells can never join the island, so cross it out. And when every path between two kept cells squeezes through one particular cell, that cell is a bridge: it must stay, or the island would split in two. Let arithmetic and shape answer each other.
Use the colors (region boards)
On boards with colored regions, each region is simply one more line with its own target, so all the tricks above apply to it. Regions are small, so they are often the fastest place to find a forced cell; check them whenever a row or column stalls.
When you are truly stuck
Pick a promising cell and play out "what if it stays" in your head, using the rules above. If the consequences crash into a contradiction (a line that can no longer hit its target), the assumption was wrong and the cell must go. Use this sparingly: almost every board falls to the simpler tricks, and the hint button will always show you the next logical step if you want a nudge instead of a spoiler.