When my 1999 Chevy Suburban is cold, I start it up and when the accelerator is depressed, the engine bogs down, and it back fires and stalls. When driving, I have to let off the accelerator and depress it very slowly and increase speed very gradually until the engine warms up....then it seems to function properly. Does this sound like a fuel pump or a fuel sensor?

SOURCE: Intermittent power loss after accelerating hard

Sounds like a fuel delivery problem! Try filter first (cheapest), then the pump
Hope this helps, Henry

SOURCE: 92 Chevy Suburban Electrical Fuel System

Check fuel flter at fuel pick-up intank (looks like a small wire sock). Remove fuel filter and fitting at fuel rail. Blow out fuel lines w/about 35lbs air press. Ck electrical @ relay - should be 4 connectors - one is key on hot/one is always hot/one is ground and last is hot to fuel pump. Good Luck

SOURCE: engine misses when accelerating DOWN THE RD

coil packs

SOURCE: 91 XJ6 Stalls occasionally in Drive with brake depressed

Idle valve? If stuck in hot position (they often do), then not enough air when engine is cold. Symptom would be that it does this more when cold. The brake pedal thing gets me a bit, does it do it when the handbrake is pulled on but in Drive? If so, nothing to do with brakes, but just that there is a load on the engine caused by halting the torque converter creep, which goes back to idle control problem. If it is only with the foot brake, then I would suspect vacuum leak, there is a whole host of vacuum circuits on your Jaguar, have a look for hoses that have rubbed on the engine and have a slight hole or ones that have perished. Pressing the brake causes a reduction in vacuum to all the circuits, but if there is an air leak, then this can cause the inlet manifold vacuum to collapse and the engine to stall.

SOURCE: 1988 chevy corsica bogs down when you press the

Bogging can be caused by several things but the most common is lack of fuel. A plugged fuel filter would be the first check. Next would be a possible fuel pump problem. Also a plugged exhaust will give the same symptom, catalytic converters the prime suspect.
If your car is using excessive oil or has been running poorly for a while will contribute to catalytic failure.