the 737 is a lot tamer than a piston trainer
i don't really find that - piston trainers don't have a swept wing, are slow by comparison, have the inertia of a crisp packet and have instant engine and control response. a jet basically just needs more finesse and anticipation, particularly on approach, but all perfectly do-able...
when you're flying raw data approaches, remember that its not really true raw data if you're flying a modern aircraft - you still have a helping hand from the track line data on the ND (even if you're using ILS mode rather than the normal moving map mode). Using that will help you anticipate localiser movement as you descend and the wind starts backing or becoming more turbulent. scan the VSI as well as pitch attitude to make sure your pitch corrections to track the glideslope are smooth and not over-cooked - i.e. don't needle chase, laterally or vertically (standard stuff from your piston training but even more important at higher approach speeds). remember the slow spool up on your engine(s). for instance, if you're slowing to bug speed, anticipate the power increase which will be required. scan N1 more often at this point to achieve this. the fan takes a long time to wind up from approach idle.
as for the v1 cuts on departure, let the plane react long enough to be certain which engine has failed (doesn't take long), dont panic and do the wrong thing. then feed in the rudder smoothly but with enough determination to stop the thing heading towards the grass, which it will rapidly if you're too tentative. don't pump the rudder - in a light twin with a straight wing you can get away with murder, but in a jet you'll do aerobatics, so smoothly does it.
as STDBYRUD says, talk to all your instructors about it - they'll have different points of view, you'll work out which way of thinking about things makes sense to you.
like most aspects of a type rating course, its the fact that time is very limited and precious in a sim which piles the pressure on. you can never practice something as many times as you'd ideally like to in order to really nail it down.
good luck with it