The push-up is one the basic exercises that you can do either at the gym or at home. No weight s are required since one is essentially "pushing" or in some instances "pulling" one's body weight.
It is important that push-ups are done correctly in order to not only be effective but also worthwhile. If done incorrectly there will not be any gains and at the end it will frustrate you.
There are many variations of the push-up; there is the regular push-up, the close hand push-up or "diamond push-up", there is the elevated push-up, etc. pick one and just go with it.