Isn't this fun? When my son hit this stage, I thought I was going to hit the roof. Especially in public. It seemed like my little angel turned into a whiny fussy monster almost overnight. The good news is that this is a stage. The bad news is that it is NOT a short-lived stage. My son is on the tail end of it, mostly not whining or crying to get what he wants, and I think he's been going through it for about 8 months now?
I tend to ignore my son's tantrums. I tell him I can't understand him while he's screaming, and walk away. If it gets really bad, we have a "calm-down spot" where he goes to sit until he calms down. This is not time-out. It is not punishment, there is no apologizing (unless he hits or kicks or something). He can get up the moment his fit is over, he doesn't have to wait for me or anything, but he has to sit there while crying until he calms down. Lately (my son is 2 and a half) I have discovered that all I have to say is, "Do you need to go sit in your calm-down spot for a while?" and he stops crying.
I also pick my battles - I say no if I mean it, but I've tried to remember that I don't need to fight over every point. I don't need to fight with him if he asks for a healthy snack, even if dinner is in an hour, because it doesn't really hurt him to have carrot sticks now even if he "spoils his appetite" for dinner. Giving in sometimes and letting him make some decisions in his life gives him the feeling of control that he is trying to get with the tantrums, and so I try to consider my answer before I say no, and make sure that I actually care whether he does what he's asking or not.
For the sippy cup, consider limiting where he can have it (like a paci). My son is allowed to have his paci in his bed. That's it. Never outside of it. Some days, when he's sick or over-stimulated, he asks to sit in his bed for a few minutes sucking to calm down, and that's fine, but it happens less and less frequently as he gets older. The way you train this is you put the sippy cup in his bed (or wherever he seems to need it most) and remind him over and over and over again that it has to stay there (not cross, just matter of fact). If he cries for it, he can have it - in his bed. Put him in, and when he wants to get up again, tell him that he has to put the cup down in order to get up. It may take a couple of days to get him to understand the new rules, but as long as he can have the cup whenever he wants (as long as he meets your conditions), I have found with my kids that they adjust very quickly. For a while, he may ask to get into bed to have it regularly, but as life outside of bed becomes more interesting, you will most likely find that happens less and less.
Also, serve drinks at meal times in big-boy cups. No sippy, no spout (hard or soft). He'll spill a couple of times, but if you make a big deal about it, he'll be very proud that he's drinking out of a big cup.
Finally, this may get worse when you give birth, even if you get it under control now. The older sibling often regresses when a younger sibling is born. It's okay. The biggest point is to remain calm and in control and not to react to his tantrums, while giving LOTS of positive reinforcement when he behaves so that he realizes that the way to get your attention is to behave. However, he's two. You may not have this under control by the time the baby comes - and that's normal - so make sure you learn how to cope for yourself. He'll get over this eventually. Good luck!