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If you don't want collisions to be able to push your objects, you can make them Kinematic (Under Rigidbody 2D -> Body Type -> Kinematic)
This tells the Physics engine "I will take responsibility for this object's movement, including dealing with collisions, so you don't have to push this object around to solve penetration"
The trick is that then you ...
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It doesn't matter that much for collisions with static objects. It gets more relevant when you have collisions between moving entities.
Personally I prefer to check first and move when the destination is valid, because it prevents two problems:
You don't have to remember where the entity moved from
You never have an entity in an illegal position
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I managed to solve collisions problem by splitting the collision handling into horizontal and vertical components, one at a time. The implemented idea is:
Move player on horizontal axis.
Check for collisions.
Solve collision on horizontal axis.
Move player on vertical axis.
Check for collisions.
Solve collisions on vertical axis.
This system does not ...
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