Hi H.,
My girls are 5.5 and almost 3, so I've been through my fair share of major tantrums. What you've seen so far has been the calm before the storm (sorry =). Kids get SO much more talented at throwing tantrums after they turn 2. It's very normal - they feel the need to be more and more independent, but their verbal and physical skills don't yet match up to what their brain is telling them to do. So they get frustrated and throw a tantrum.
Tantrums are the child's way of expressing frustration - not even about what has JUST happened, but maybe a combination of many things.
With both of my daughters, what worked for me was to very calmly lead them (or carry them if they are in writhing-on-the-ground mode) into their room and tell them, "You can come out when you're done." Usually if they do not have an audience, the tantrum will play itself out faster. I've found that by focusing on or reacting to the tantrum, it makes the tantrum worse. Your child already feels out of control, and what they need is for you to be calm and in control of the situation. By giving them a safe place to have the tantrum (their room), and giving them permission to feel the way they do (although out of your immediate earshot), they will feel secure, yet still a little independent.
Take heart, as they approach 3 the tantrums decline quite a bit! =)
Hang in there and good luck.