The simplest type of animation, moving things around and fading in and out, can be done with a few static methods of UIVIew. You can affect the center, bounds, transform matrix and alpha level of one or more views.
[UIView beginAnimations:nil context:nil];
[fadingOutView setAlpha:0.0];
[slidingView setCenter:CGPointZero];
[shrinkingView setFrame:CGRectZero];
[fadingInView setAlpha:1.0];
[spinningView setTransform:CGAffineTransformMakeRotation( M_PI )];
[UIView commitAnimations];
Animations start with the current state of the view and interpolate to the state assigned between begin and commit animation. So if fadingInView already had an alpha of 1.0 (the default) there would be no change.
If you are unfamiliar with static methods [UIView method];
means call method on the class not an instance.
Using other UIView static methods you can control several details of the animation. Every UIView has a CALayer that also has a few properties that can be animated, the most interesting of which is the 3D transform property.
If the basic animation is not sufficient for you needs, you can either look into CAAnimation and related classes, or look to a third party animation library.
I think the best place to start learning is in your code, since you are just transitioning from Flash. Look at the very bottom of UIView.h to see the animation methods. Make a few views and move them around.