If I push on a table with some force 'f' then by newtons third law of motion the table also applies the same magnitude of force on me in the opposite direction.
This is a true statement.
I get that there may be various forces that would be acting on me but then those forces (like friction) would also be acting on the table
This is not a true statement. Newton's third law doesn't say "If I push on an object, then all forces acting on me must act on that object as well." If you push on the table, then the only force acting on the table related to you is that push (ignoring negligable gravitational attraction). If there is friction involved, then the friction force on you is not related to the friction force on the table.
why is it that I don't move but the table does?
This is due to friction. If the friction force between you and the floor is strong enough to stop you from sliding, then you will not move. If friction is not strong enough to stop you from sliding, you will move. The same is true for the table.
In fact, if you pushed on the table with both you and the table on a frictionless surface, both you and the table would move.
Essentially any of the four scenarios are possible (you move/don't move and table moves/doesn't move) depending on the applied force and the friction that is present between the various surfaces in question.