My advice is to watch The Dog Whisperer. Of course, your children aren't dogs and you can't SSSH them or yank their collar to change their behavior. But, there are several things from that show that I have found to be very helpful with my kids.
Firs off, recognize the tantrums as "protest behavior" - the kids are just protesting the rule. Sometimes protest behavior can be ignored and sometimes it must be addressed. Be sure to have some consequences in your mind in advance so that you set boundaries you can and will enforce.
Calm assertive is SO helpful in my house. Sure, I blow my top now and then... but we're only human. I'll let Cesar address this for you, but I find it VERY helpful.
One other Dog Whisperer thing I like is addressing the behavior early, before it escalates. For example, you see your kid walking near something you don't want them to touch. Instead of waiting to see if they touch it, remind them right away. I've found this VERY helpful in public.
Some people will tell you that you must get them on schedule. I know lots of parents that live and die by an exact bedtime, an exact lunchtime, etc. I do think it works very well for them. I do think kids like it. I don't have one, though, and I have great kids so I don't think it's the only recipe for being a good parent. What you lack in time structure, you can create with 'rituals' so that the kids understand what order things happen. So brush your teeth, read 2 books, rock and go to sleep. Or wash your hands, pray, and then sit respectfully and eat. We eat at a variety of times. I do make sure I have fruit or something to 'hold them over' if we're eating later. Lack of food or sleep is a sure-fire temper tantrum over something else...
I firmly believe that kids understand far more than they can communicate. So I talk to my kids as if they DO understand and only backtrack when they ask or get that quizzical look on their face.
Sharing: Two minute turtes has worked well for us. If a toy is disputed, set a timer and each kid gets to play with it for 2 minutes. If they can't do that, the toy is put in 'time out.' If only one kid is the 'problem' - he/she sits down in a time out type situation during the two minutes the other kid plays with it.