The method I use uses special 'cutscene missions' - missions entirely dedicated to being a cutscene. My personal design ethos shies away from cutting to a movie mid-mission; I use cutscenes to show things the player won't see, rather than the things he is already.
To follow this idea, you need to first set the mission up so it doesn't have a briefing or debriefing - look at the mission specs for that.
Then you need to turn Alpha into a camera. Set every subsystem to 0, so the player cannot move, turn, fire weapons, cycle through targets... nothing. Protect it and invulnerabilise it for good measure.
Create a bunch of ships - fighters, mind - and use FRED's 'use current ship as camera' option to move the various ships to all the different camera angles you want to have. Create a waypoint for each ship and put it in the centre of the screen view you get when using each ship.
Basically, what will happen is you'll move the player ship to each of these positions in turn as the cutscene progresses. Using set-ship-position, you can, for example, start off looking at cruisers speeding towards you, then switch to a different position watching them go by from above, then get a nice shot of them from behind. You need the waypoints to orient the player ship each time - else he'll keep pointing in the same direction all the time, and the player'll be looking off into space when he's supposed to have turned around to see all the action.
So: note the co-ordinates of the various ships and their corresponding waypoints. Probably advisable to sort them into the order that you want to go through, else you might get confused and screw it up.
Create an event for each camera change. Use the sexps set-ship-position and set-ship-facing, using the X, Y and Z values of the ships and their waypoints, in the events, and set the event to trigger at an appropriate time. When the first event triggers, the player will be moved to the first event's ship position and be turned to look at the first event's waypoint position. When the second event triggers, the player will be moved to the second ship position and look at the second waypoint, and so on, and so forth.
At the same time you will obviously have created something for the player to look at.
You can use hide-hud-except-messages to, well, hide the hud apart from messages. I like using the training message window to convey information, because it's different and everything. I also understand that there are some new sexps that let you fiddle around with the camera itself (instead of the ship) and have all sorts of neat effects... I haven't looked at them yet. However, the above method produces admirable cutscenes, I feel.
Don't forget to delete those extra camera ships once you're done with them.
