First of all welcome!

I keep all my soaps in the containers they came in and I have a separate mug that I lather in.

The best thing to check is the lathering article in the Wiki here and follow the instructions there.

Some people prefer to put their soaps in a mug, but its really personal choice. Sometimes the puck fits just right, other times you need to make it fit. The easiest way to do that is grate as much of the soap up as you need to fill the container to the level you want, and then press it into the bottom of the new holder with some water to make it all stick.

If your soap is in a mug or something, I'd just leave any remaining lather there as it will form part of the next lather you make.

If you make your lather separately in a different mug, its up to you what you do. I try to make only as much as I need and I rinse away whats left (usually not more than a small smear as I'm pretty good at judging the amount to make now!) as I use different soaps each time.

Good luck and keep us posted!