Mountain Climbers are HARD! You better believe that they kick my butt every single time I do them, and I have been doing them regularly for the past few months.
Goal for beginners: GO SLOW! Don't worry about speed in the beginning. But correct form. Keeping your butt down as much as you can.
Also try to keep the weight in your heels so it takes pressure off your wrists.
And step your foot in like in the picture here instead of a more advanced version of bringing your knee to the chest and your toes stay off the ground. Focus on the basics and then you can make it more difficult when your body is ready.
Don't get frustrated if you do a few and are already thinking about stopping. Aim for a small number and then increase that number the next time. Start with 5 then 10, then 12. And so forth. Great goal would be to do mountain climbers for 1 whole minute!
Remember, you can do this! I promise you! It is only for just a short time that your body will fight you doing this but once you are done, you are done!