Once, while I was in B.C. about to leave for Desolation Sound, I looked in the engine room and there was water spraying. Good thing I checked. I shut down the engines and replaced the offending hose. After you leave the marina (or an anchorage), wait until temperatures come up before slowly bringing the boat to cruising speed. Watch your temperature gauges(s) closely for your first few minutes. Make sure the engines come up to temperature and then stop rising.
Before heading back to our marina, I run the engines up to full speed, checking to see if everything is working and that, in our case, both engines come up to the proper and same rpm. I also check the speed and feel for vibrations. This is not a long full-speed run — just a couple of minutes. Leave enough time to slowly back down and then idle to the marina long enough for the engines to properly cool down before entering the marina (or into the anchorage). It is much better for the engines to cool down with a bit of pressure beyond just an idle.
A good piece of general advice is be kind to your neighbors and quickly get started and away from the dock (or out of the anchorage). At the dock, get all your lines ready for a quick departure. Unhook power cords and cable TV lines so that once you start the engines and do all the checkouts, you can leave. Engines do not need to reach temperature before idling your boat away. Certainly, there are exceptions, such as if soon after leaving you will face heavy currents, rough water or heavy winds. Likewise, when you come into a marina or an anchorage, be thinking of how quickly you can get tied up or anchored and shut things down.