There must be a few "must-sees" among your 'shows and plays', so it might be as well to try and book them in advance if they're important to you. Doesn't mean you can't get discount, but at least you get peace of mind. That's how I tend to approach it when I visit NYC.
My favourite one-stop-shop is theatremonkey.com; special offers, seat reviews, show descriptions etc. Broadwaybox.com/london is also worth a look for discounts. I've had some great bargains via lastminute.com, but the only downside is you don't get to find out your seat allocation until you pick up your tickets at the Box Office.
Off the top of my head, Sister Act, Hairspray and Wicked all offer "day seats" - £25 front row tickets which you can only buy on-day by queueing at the box office. I'm sure someone else will chip in if there are others I've omitted.
So what are you hoping to see?
Seen some shows in my time....