Recently I changed the oil in my car though I think that is incidental and not related to the problem. Not long after I noticed that my car, if accelerating from a full stop is shaking as if it will stall out but after I get out of 1st and 2nd gear around 40 mph it drives fine. But whenever I make a full stop and get going again it does the shaking thing. What could the problem be?
WAG, you’ve got a bad plug, or two, or bad plug wires. Could also be a vacuum leak.
So is this problem on your 1962 Borgward or your 06 Rolls Canhardly? 
I can think of several things that might be an issue, but depending on the year make or model, some would apply and others would not.
97 Toyota Celica. Like I said after I get out of first and second gear it stops shaking and it also does not shake at all in reverse, I also noted an odd odor when I used the AC.