I am just starting and am feeling dizzy already now I know why, that suggestion of resting for 20minutes seems perfect, because working so hard without rest, wear off energy in our body tissues, ligaments, tendons and leave us with so much stress shown on our faces and body especially for new members like me, so I make adjustments until I find a perfect solution that will propel me to do push up better everyday.
Most times eating light food such as fresh or frozen fruits and vegetables prio to your pushups will help in keeping you active and not being dizy. However it is also important you make some body movements, stretch your body or probably take a few health walk after then , have a target of range of push ups you will be doing for few days without any intentions of hitting above your limit while your body endure the pains that will enable you gain enough stamina to break your limit, be consistence and give the body rest when needed and you can't imagine how much of progress you will be making with time.