The basic plank hold and its variations are simply the best core stabilization and strength exercises you can start with!
Upper push up position starts with the shoulders and wrists aligned, abs braced, ass squeezed and neither too high nor too low.
I like to think of the floor pushing against me and me pushing back against the floor with my fingers spread. I also reach through my heels ;)
The mountain climber is also a great core exercise! It is so versatile that it can be used as a metabolic exercise as well :0 In the photos I am performing mt. climbers on a slide board.
A slide board is a great tool to have on hand. If you have wooden floors you can put hand towels underneath your feet, or use a valside, if you have a carpeted area.
Instead of bumping up and down with your bum in the air, the slide board mt. climber forces you to really stabilize and engage your lower abs as you bring your knees into your chest one at a time.
Once you perfect this move you can bring your knee to opposite elbow or bring both knees into a pike position :)