Step 1: Desoldering
First bend a piece of thick copper (or other solderable metal) wire to fit around the component, and contact all its solder pads. Solder it in place. This is done to make thermal contact.
Then heat the wire at a central place with your soldering iron, and lift the component away with tweezers. Or push it away using a wooden toothpick. Only don't be too enthusiastic with the push or it will fly across the room and be lost forever in the crud that covers the (in my case anyway) floor.
These are best stored inside foam cutouts in a suitcase. Miniature, of course.
Step 2: Tantalum capacitors
I'm sorry, I do not have further info on identifying them other than checking them in a multimeter which has a capacitance range.
They usually look like little bricks with metal strips issuing from their ends, folded over the bottom.
On the left of the picture are ceramic capacitors - MLCs - short for Multi Layer Ceramic - usually they are this brown colour, though I have seen black and white, and the ends look as if dipped in metal. There are usually no markings at all, so measurement is the only way to classify them.