Why does the airplane make my stomach "go up"?

Gravity acting on your stomach. There is a membrane in the mid-torso which supports the stomach and is flexible. This has a certain amount of 'give' which allows it to shield the stomach from undue shocks and stresses. When the aircraft climbs, the whole body starts moving upwards. The membrane flexes and allows the stomach to lag behind momentarily before letting it catch up with the body. This is the feeling you describe. This feeling is also pronounced when descending and especially in steep dives.