I'm trying to gradually slow something down without losing audio sync at the beginning of the clip. The part that I want to gradually slow down does not have audio, so there is no need for sync there, but it is tied in with footage that cannot change in the beginning.
The key to understanding Time Remapping is knowing that it's like any other keyframeable property: you have a value changing over time, and in this case, the value is the displayed frame of footage.
This means that the value change (playback speed) must be the same as it was (1 frame per frame) when you first chose Time Remapping (TR) if you want to keep audio sync.
If you want to keep sync for the first 10 seconds of footage, make a TR keyframe at the end of the first 10 seconds. You now have three keyframes for TR. Then you should drag out the last keyframe to make the footage slow down at the end. R-click on the last keyframe and select "keyframe assistant>easy ease in", and do the same for the middle keyframe but select "easy ease out".
Open the graph editor and make sure the TR graph's speed is continuous by dragging handles.
[Dustin Ward] "The part that I want to gradually slow down does not have audio, but it is tied in with footage that cannot change in the beginning."
You can also split a layer and just time remap the part without audio. This probably isn't necessary if you can get what you want that way, but you can also duplicate a layer and make one copy just the video and the other just the audio ... if you do that, just don't forget you've done it and move them independently in the timeline.