Exactly right. A nice fresh coat of oil won't be your friend this time. Before you place the taper in the tailstock (or spindle, as the case may be, wipe out the inside socket with a clean, dry cloth. This will remove any particles that might be there. It's a good idea to take a look inside with a flashlight and check for any dings or junk that might be in there.
Wipe off the tapered shank, then slide the chuck, bit, bearing centre, or whatever about half-way in. Then slide it briskly home. You want just enough force to make it stick well.
In the tailstock, it's a good idea to crank the tailstock out far enough to make sure the end of the taper won't hit the solid end of the socket. Most tailstocks release the tool when you crank the barrel back in all the way.
To release the taper from a spindle, it usually requires inserting a bar of some type in from the outboard end of the spindle, then giving the bar a sharp rap or ten with a mallet. If your spindle tooling is held in with a drawbar, leave the threads engaged but loosened. This will keep the tool from sailing across the room when you succeed. :yikes: If you're using a solid bar (I use a piece of broomstick for this), place a large rag over the tooling to cushion it and the lathe bed when it takes flight.