I went back and read the original post again, just to make sure I hadn't missed something important. OP appears to say that when the car is in first gear (automatic trans), the car will rev, lag, rev and then drop into gear and go forward. Did I miss anything?
Unknowns: mileage on the car, transmission fluid level, fluid and filter condition (recently serviced), does the OP get an audible 'clunk' or noise when it goes into gear. Does the problem occur in D or only in 1st.
Thoughts: If fluid is okay (appropriate level and not burnt), then there is an issue with the trans. While there would be a small amount of difference (lag) from pressing the accelerator to the car going forward, the symptoms the OP describes do not appear to be normal. Before an accurate diagnosis can be made, the OP should have someone ride along with a scanner attached to see what the trans is doing. With that information and answering the questions about fluid level and condition, we might be able to better help. Just my
