There have been some great recent examples of goals coming off of the forecheck where teams are quick to get initial pressure on the puck, get 2nd quick support and immediately find the release to the strong side dot area for quick strike attack
One of my favourite sound bites lately was from Jared Bednar describing the forecheck as the FIRST indicator of effort and compete. This is evident in these clips, followed by supporting and playing fast to the strong-side dot while the opposition is disorganized

A favourite practice activity of mine to work on this is the 3v4 Forecheck/Breakout game ("stolen" from a Dan Muse presentation) The puck is placed into open space for the four breakout players to retrieve, execute a breakout, and attack in transition. Meanwhile, the three forechecking players attempt to create a turnover and counter attack
To emphasize the quick-strike principle and playing fast off changes of possession, the forechecking team is awarded bonus points for shots within “X” seconds of a turnover, and triple points for goals scored within that same time frame. Goals scored after the time window are worth one point. Conversely, the breakout team earns extra points for completing clean zone exits leading to chances or goals within a designated time frame
There are many ways to manipulate the constraints to target specific concepts. One variation I’ve found valuable is allowing the forechecking team to place the puck themselves, instead of the coach. This makes the retrieval far less predictable for the breakout group and forces the forecheckers to work on strategic puck placement. To add even more pressure/game specificity, the breakout players round the net and pressure the puck carrier who is placing the puck