2001 Chevy Suburban Car is slow to accelerate from a standi

First you may need to provide an engine size, is it a 2500 or 1500 series, and if you know which trans you have. It sounds like you have already determined it is starting in a gear other than drive range, first. What you are accomplishing when selecting manual 1st, is significantly different than drive range, first. Manual first is accomplished with the addition of the lo/rev clutch, and overrun clutch. The question is, can the actual condition be described as a second gear start, second/third gear only, or the like? To answer your question, it is a transmission internal issue. At this time, that is about as specific as I can be. Have you checked the fluid level, and it's condition(color/odor/burnt)?