Hi Isaac,
You need to accelerate to achieve sufficient airspeed to create enough lift to fly. The acceleration is opposed mostly by the aircraft inertia (mass). The 2 components of Drag (friction and lift-induced) only become relevant with speed and lift (itself a function of speed).
Hence, as you accelerate along the runway for take off, Drag is not the main force you are overcoming. A small increase in drag - or reduction in L/D - is acceptable at that stage if the gain in Lift Coefficient is sufficient; since you can lift off at a lower speed your ground roll is shorter.
But you will climb slower with flaps extended than 'clean'.
On some aircraft the gain in lift coefficient is so small it doesn't merit using flaps for takeoff. This is the case with some (most? all?) C172 models.
Using flaps for landing also is not necessarily due to the extra lift, but because (a) the change in aircraft attitude means you can see better out the front and (b) you may want to slow down.
B.