So I'm not claiming AT ALL to be an expert on this (since I've just arrived myself), but if you're in the same situation, let's commiserate. I'll tell you the ways I'm trying to cope, and they may be all wrong, but a girl's gotta start somewhere...
Ignore the Tantrum
The tantrum, not the baby. But thou shall not fear the tantrum, give in to the tantrum or try to match the tantrum. I've been trying to just stay calm (or at least look like it in her eyes), hold her (as she kicks and screams), remove her from the situation and do anything else with her calmly (like reading a book). It might not always soothe her, but it does wonders for me.
Make Her/Him Laugh
I've noticed that if I spend more time being silly with her rather than scolding her, she's a lot more likely to want to please me. Now that gets REALLY tricky when the silly thing she wants to do is wrong. Then, that brings us to what I just discovered...
The Power of the Gentle "No"
Whenever I tried to sound extra angry when saying "no" so she'd get my message, Melissa thought it was hilarious. She'd even run to me giggling for a hug, and then go right back to doing what's wrong. But when I just said "no" in a serious but not as entertaining way (and, of course, led her to something else), she was more likely to follow.
Notice that I say "more likely", which usually means on a good day, when the sun is shining, her belly is full and she's well rested. And even then, sometimes (often) the meltdown is coming, and there's nothing you can do about it. Welcome to toddlerhood.