OK. Just had lesson number two. Now we're looking at a Reverend Gary Davis/Mississippi Fred McDowell number - "You Gotta Move". Great fun. I like this piece because you can play it quite briskly, or play it real slow and turn it into a lament. (well... you can, I need some practice).
We talked a little about different kinds of slides. Here's some considerations about that tapered slide....
1) It means you have more metal on the heavy strings, a little less on the light strings, so some players find that gives them a balance that they prefer when playing chords.
2) If you think about vibrato with the slide, this involves a fanning with the wrist. The tapered slide helps to even out the vibrato across the strings without changing the wrist movement as much from 1st string to 6th.
Ha! And I now have a THIRD slide

This is a Blues Bottle (that's the trade name). It's not tapered. The shot is just at an angle.
I find this has an advantage in one respect in that you have to bend the finger a little rather than just slide it over the finger. This helps control by making it a little easier to regulate the downwards pressure - is what I think right now.
Lots of fun. Lots to learn.