Most obviously: If you have a grip -- usually with a hand -- on your partner or your partner has a grip on you (or both have a grip on each other) you can use the connection to pull or push. Note that if your partner has the grip they may elect to release, which is why in jiu-jitsu when grabbed, we often grab back (so that they can't get away).
In principle it is possible to grip with your toes or your teeth, but I will disregard those possibilities for now.
Use of the hands
When gripping, especially when gripping clothing, it is good practice to grip tightly with smallest two fingers (the ring finger and the pinkie), and quite gently with the thumb and the remaining fingers. This prevents over-tensing, increases sensitivity to your partner's movements, and over time strengthens the smaller fingers. This is also the recommended way to grip a sword.
Usually -- there are exceptions -- the shoulders should be relaxed and lowered to allow power to be transmitted more effectively from the core of the body. Elbows may be raised or lowered depending on the situation.
Other ways to push
In the absence of a grip one can push with almost any surface of the body. Fingers, palms, wrists, forearms, elbows, shoulders, head, chin, chest, stomach, side, hips, buttocks, knees, shins and feet can all be brought into play.
A push can be used to project away or deflect. Of course many a push can be transformed into strikes, but that's another subject.
Other ways to pull
Besides gripping it is possible to pull or draw-in wherever a sufficient angle can be created at a joint. Hooking and trapping describe many of these kinds of actions.
Pushing and pulling at the same time
By using push and pull together we can generate more complex twisting and turning actions. These are more difficult for an opponent to interpret and neutralize than just using push or pull alone.
On a large scale these two-way actions are essential to most throws: We make large circles. On a small scale they contribute to effective joint-locking: We make small circles. Speaking of small circles, here's a taste of Small Circle Jiu-Jitsu founder, Professor Wally Jay (aged 70 at the time of the video):
In his method Professor Jay emphasizes a wrist action similar to how you would turn off a tap; it combines push and pull. You can see the students doing exercises to practice this action, and Professor Jay putting it into effect in many and varied techniques.
There are many, many ways to push and pull. As you can see from the video, push and pull works with sensitivity, timing and flow to produce excellent technique. This is something that you can reasonably aspire to still be doing at age 70 (and beyond).