Pre-draft: Study your leagues scoring system. Some players look a lot better depending on scoring. If you are in a points league where blocks/hits are not scored players like Callahan, Backes, etc. are not going to be as valuable. Especially check your goaltender stat values. If stats like saves and games played are scored then goalies like Lundqvist are going to be most valuable due to the amount of net time they see. If values for wins, losses (negative value) and shutouts are much higher than goals/assists then it will be beneficial to make sure you waste an early pick getting at least one good goaltender.
Head2Head and Rotisserie scoring are played fairly differently than points based scoring. You will be aiming to win multiple categories every night and you need players who are jacks of all trades, rather than a guy who can pot 80-100 points, but has a terrible +/-, low faceoff wins, and no hitting. Again it all depends on what stats are scored. Generally you can depend on guys who are hitting 90+ points in a season to be pretty well rounded in most categories, but do some homework.
Biggest mistake: Taking a young/fringe player because they look good in pre-season or are expected to produce. If you want players like Hall/Eberle/Seguin get them in late rounds. Don't fill your roster with them.
Also do not pick players just because they're on your favorite team (assuming NYR) or conversely avoiding players because you dislike them. Some players you may hate are fantasy gods. (Crosby, Broduer)
The most important thing is to follow as many teams/players as you possibly can. Getting a hot player from the FA pool and riding him before someone else does can make a huge difference by the end of the season. Especially in leagues where there are no max moves, and no max games per position. Riding hot hands should be employed without a doubt in those situations.
Lastly, if it ain't broke, don't fix it. Don't drop/trade players who has one bad game, or slump for a month, unless they were hot hands you picked up to ride previously. Most leagues won't let you drop certain players, but it's never a good idea to send a guy you wasted a 1st round pick on to waivers. If you are making trades, buy low, sell high, obviously. If someone on a top line gets injured, don't just think about replacing that player, also consider the effect it might have on the rest of the line.
For example last year, after Smyth got injured, Kopitar's production slowed down a bit. So anyone who thought that might happen, and sold Kopitar high probably got a good return on him. Holding on to him was fine as well, don't get me wrong he had a great season, but sometimes thinking ahead can be the difference in good/bad moves.
That's all I have.