firmly, hard
You'll have to push harder if you want it to move.

angrily, roughly, violently
gently, lightly, playfully
carefully
deliberately
He was deliberately pushed into the path of the vehicle.

blindly, frantically
hastily, hurriedly, quickly
slowly
suddenly
just, merely, simply
Jack flung himself at Steve, but he simply pushed him away.

aside, away, back, down, over, together, etc.
She leaned on the box and pushed it aside.
They pushed the two desks together.

try to
manage to
begin to

against
The fans pushed against the barrier.

at
She pushed at the door but it wouldn't budge.

out of
He pushed her roughly out of the door.

through
I tried to push through the crowd.

to
The woman had been pushed violently to the ground.

towards/toward, under
She found a note pushed under the door.

push sth open, push sth shut
He managed to push the window open a few inches.

aggressively, hard
Lucy should push herself a little harder.

too far
Her parents are very tolerant, but sometimes she pushes them too far.

around
Don't allow yourself to be pushed around by that bully.

into
Her parents pushed her into accepting the job.

for
The two governments are pushing for economic reform in the region.

push sb/yourself to the limit
He felt he was being pushed to the limit of his self-control.

Push is used with these nouns as the subject:
hand

Push is used with these nouns as the object:
agenda, baby carriage, bang, bicycle, bike, boundary, buggy, button, cart, earnings, fringe, handle, idea, index, initiative, knife, legislation, lever, limit, luck, needle, pedal, pram, pushchair, sale, spectacles, strand, stroller, thought, trolley, way, wheelchair

big, hard
gentle, little, slight

give sb/sth
She gave him a gentle push.

at the push of a button
The machine washes and dries at the push of a button.

big, extra, major, strong
renewed
final

make
The company is making a strong push to expand its distribution.

push against
the final push against the enemy

push for
There has been a big push for higher standards in schools.

push towards/toward
a push towards/toward organic food