Check the engine and tranny mounts. Sounds like the engine rocks to the side slightly when you apply power. First gear's more pronounced due to the torque multiplication.
You may be able to see it rocking - hood open, watch the engine, have someone move it in first gear.
First look at them, you may see they are deteriorated.
Then jack up the engine enough to take the weight off it while watching the mounts. You'll see them try to separate if bad.
Do the same for the trans mount.