best way to do puzzles
This is the best recommendation for solving puzzles fast.
Checks, Captures Threats.
When you start a puzzle look for checks. This is a good way to start because a series of checks might end up checkmating.
Captures are a good way to start your puzzles if there are no good checks on the board. After all you get material when you capture!
Threats
If both fail, then the last option you have is threats. You could threaten a unstoppable mate or get an advantage!
Hope this helps!!!
In chess classes, we weren’t allowed to use the board because it trains us what to do in a real life situation
It's certainly better to solve puzzles without moving pieces. Visualization is a super important skill which you do not train when you solve puzzles by moving pieces.
That is not to say that you should always avoid it when analyzing positions, because when you move pieces, you can analyze much deeper, which is sometimes what you want (when you analyze your games, for example).