there must be an initial acceleration when the aircraft enters a climb from S&L.
To cause that change requires unbalanced forces. It could be done by increasing power or changing angle of attack (or a combination).
This unbalance changes the velocity of the aircraft (ie causes an acceleration) from a level direction to one angled upwards.
In the 'steady' climb that (hopefully) results* after this change all the forces will again end up balanced, with the magnitude of the lift force being less than weight, hence my original statement, lift can be less than weight in some modes of flight).
*ok, in real life we cannot sustain a steady climb forever but for small climbs of a few thousand feet, we should be able to approximate reasonably closely a steady climb.