Well the good news is that the earlier into the "terrible two's" the sooner you are out of them :-)
I agree with earlier posts- learn what you can about child psychology! While every child is different and the same thing does not work for each one, the more you can arm yourself with knowledge the better you can develop the best plan for YOUR litle one!
We have an 18 month too. Some things definitely worked for us...Distraction, communication and consistancy.
For us the public tanturms are hardest (in the house no problem!), our DD is most sentitive when she is tired (some are set off by hunger or boredom)so if we had a bad nap day - we probably aren't hitting the mall late that afternoon. We try to keep realistic expectations and keep our plans flexible. When out, we keep a "distraction kit" in the diaperbag (stickers kept in old greeting card envelopes, a book, etc). If we don't get to the last errand and head to the park instead to blow off energy- no biggie.
We have tried to help our DD communicate better with signing. Invaluable! She was big on whining and growling early on so the signing helped her communicate. we are big on positive reinforcement and lots of praise when she is being good..she now tells us "good girl" when she is being good (she's getting it).
Finally, consistancy (you may not see it now - hang in there!) will pay off. no means no...there will be flip outs. But if you waiver she will know and then it is a battle of the wills and trust me she is not at all embarassed to wig out in the middle of Target :-). Cinsistency also needs a united front. My DD was flipping out when she could not play with my cell phone...i later found out my DH was giving her his to play with. no wonder!
They certainly know the difference between yes and no but at this age, it is irrelavant to them. A tantrum is just a tantrum...it will happen but it will pass. It's not a reflection on you as a parent!
Good luck!