With the exception of cruise missiles, most missiles don't have any obvious lifting device.
If you watch a missile being fired, prior to its rocket firing it does exactly what you'd expect: momentarily plummet toward the earth. And yet the moment it ignites, it stops falling completely, but without any pitch that you'd expect to counteract gravity; even big missiles like the Phoenix or HARM seem to have gravity-defying characteristics.
Missiles that fire on the rail seems to experience no drop at all.
AIM-9M launch from an FA-18F source
There's a good slow-motion video of the firing of an AMRAAM here.
Now this isn't magic, so I presume either a) the little fins make adjustments so that the rocket is, despite appearances, pointing a little downwards or b) the fins are smart enough to configure themselves to provide the lift needed to keep the missile in the air. But it might be neither.
The question is - generally, how do conventional missiles fly?