Soak the Main Slide
The trombone slide is the 'money' part of the trombone. Being lubricated for smooth motion and then repeatedly blown into, it naturally collects the most dirt and grime. Most of this grime will deposit itself inside the the inner slide (the chrome side that connects to the bell section).
After removing the leather slide covers, I took apart my slide and carefully set it in the bath tub to soak. I avoided moving the slide around a lot - this may significantly dirty the water.
A slide that hasn't been cleaned in many months will probably have a fair volume of deposits inside. If those deposits start floating out right way, it is likely a good idea to rinse the two slide pieces thoroughly (inside and out) to remove most of the grim. Then re-fill the tub with just enough slightly warm & soapy water. Soaking is important, but it probably doesn't make great sense to soak the slide in very dirty water.
This is a good point to check that there was actually enough bath water. The slide should be just submerged. It makes good sense to fill the outer slide (the brass half) with soapy water, too.

