Since your Silverado still has reverse gear, it is obvious the problem lies with first, third or fourth gear in your automatic transmission. It may seem a bit odd to leave second out of the list but it really isn’t because second and reverse often share common linkages and if you have reverse, it is more than likely that second is still good.
The problem is that since your Suburban’s transmission won’t slip into first — and subsequent gears — that you won’t know that at least part of second is available. It’s of little comfort to you, though, is it? If there’s no first, then it really doesn’t matter if there is a second or at least part of a second.
The problem more than likely lies in the clutch packs. The first gear clutch is not engaging and because it isn’t, your truck won’t go into forward gear at all. Since first, third and fourth share many of the same linkages it stands to reason that it won’t go into those gears at all, as well, though, as noted, second is probably good — you just can’t get there from here.
Really, it doesn’t matter whether second works or not because if your truck can’t go into the forward gears, then, you can’t get to second, anyway.
Like as not, not only are the clutches for first, third and fourth shot, but it is also likely that the bands are shot for each particular gear. This means that the transmission is gone and should and should be replaced. You have two choices, purchase a rebuilt transmission or get one from a local salvage yard, provided it is in good shape and the mileage is reasonable, and get the old transmission out of your driveline.
This should cost between $2,200 and $3,800, depending on the transmission you use.