I want to record a short animation in which a simple object would totally change its form and grow into a complex form with space inside. I tried with simultaneous extruding and key framing it, but at the end it recorded just the last version of the extruded object. I would also like the camera to move in circles around the object and then jump inside it. Is it possible to do it in blender?
This is possible, but not as simple as you had hoped. There are multiple techniques for animating the shape of an object: armatures and shape keys are the two easiest ones.
It is not possible to animate the topology of the object. If the object has 200 faces and 240 vertices, it will have those same 200 faces with the same 240 vertices even if you relocate the vertices.
To work around this you can use other techniques. When the topology of the object has to change you can hide one object and unhide the replacement object that has new topology. (because the camera icon that represents renderability in the outliner can have keyframes).
Another technique is to conceal the extra topology (because it is really hard to tell that a flat surface is really made out of 100 quads until those vertices start to move out of the plane).
Alternatively your bursting spheres might just be independent objects that don't modify the original object.
It's going to vary depending on exactly what effect you want.
As you set key frames, be sure to move further down the timeline before setting another key frame. A key frame just sets the value of the properties you are choosing at the current time. If you want to make the changes "live", you can too. In the bottom right of your Timeline, there should be a red button icon that will record your changes and make them key frames as you change it. This isn't quite as advisable IMHO because it gives you less control in the long term, as it marks a new key frame every frame.
I'd advise using an emphasis on the key part of key frame (har har) by creating key frames with a start value and end value at your start frame and the frame where you want the animation/change to end.
For instance, if you want a sphere to grow in size you would set a frame at the start frame with a set value for scale, then 30 frames later set another key frame with a higher value of its scale. This would look like the sphere slowly (or quickly depending on how far apart the key frames are).
This will work for changing the location of the vertices as well, if you use edit mode and select the specific vertices you want to move and change their location. Mark a start key frame and an end animation key frame where they have moved to your desired position.
Edit based on first comment from OP:
For your use I'd probably just create a particle system if you want a bubbling effect like on a cauldron or soda. If you want to use those objects after the burst out of the main object, you can key frame the objects by right clicking the Location/Rotation or Scale slider in the Object Properties tab, and selecting "Insert Keyframes". You will want to set a keyframe at the start frame, and then keyframe another frame further on where the bubble have come out of the object.