I very recently bought my first road bike. It is my first bike ever with presta valves. I can't figure out how to use these. I push the pump head onto the valve as far as I can. I turn the lever. This is when the leaking starts. I have to position the lever about halfway between the open position and full-over to the schrader side (to use the presta side) to minimize--not stop-- the leaking. Then I pump vigorously to outrun the leak and yank the pump head off before too much air escapes.
I have become familiar with the nut lock on presta valves. I'm using a Topeak Joe Blow Sport floor pump and I have tried a Schwinn floor pump that is similar in appearance. I don't see any marking on it aside from "Schwinn." Am I missing something or do I have a defective pump or valves?

I think you're pushing the pump head too far down the valve stem, forcing the valve to stay open.
Try a more shallow engagement.

Is your pump configured for Presta.
Some modern pumps are smart and can be used on Schraeder or Presta.
Some are dual and have 2 holes.
Many are reversable and need converting to the correct valve type.
They have a rubber grommet on top of a plastic valve pin held by a large plastic retaining ring.
For Schraeder, the valve pin points to the grommet which is places wide hole outermost.
For Presta, the valve pin is reversed to point back to the pump and the grommet reversed to show narrow hole outermost.
The retaining ring is screwed back tight, but make sure the locking lever is disengaged. If the lock is ON, you can't screw the retaining ring down fully.

https://www.youtube.com/watch?v=1JK9hR_Y8Mc

This may help

I agree this might be your issue. If you look carefully at the presta valve stem, there's a threaded portion right below where the little nut you unscrew to put air in is located; that's where the valve cap that came with the tube screwed on. Then immediately under that little threaded portion is a smaller diameter smooth bit, and under that is a larger diameter part that may be threaded or not (threaded ones usually come with a nut that you can use to secure the valve to the rim).

The pump chuck/head should have an o-ring that locks onto the little smooth area to form a seal. If you have the threaded valve stem and push it on too far, it tries to seal on the threads but air can just run out through the threads since you can't form a tight seal on the threads unless you seriously deform the o-ring. And even if you do, you've chewed up the o-ring so that next time it's even more likely to leak.

Also, a presta valve only stays closed because the pressure inside the tube forces the valve shut against the lower air pressure outside. So if you push the chuck on too far such that it's pushing the little threaded pin with the nut, the valve will be stuck open while you're trying to fill the tube with air.

With the Silca bell shaped pump head , as my example, it's gasket engages the smooth band , that is machined in each stem ,

between the course thread for the cap and the finer thread for the ring nut, forms a perfect air seal ..

and is much easier to remove the pump head when there remains air-pressure in the hose..

for me this has been my solution. it's not for everyone, but it is for many ...

That's my fave !

Don't forget to remove the tire & tube before using...