Adding to the comments thus far…
TC3 Do you learn the whole piece per usual and then start using the page turning pedal only when you're ready to play through, or do you practice with it (for the relevant sections) from the get-go?
I use the page turning pedal from the very beginning. I’m not sure what you mean by “for the relevant section,” but 99% of my pieces are more than a page long, so I’m pretty much using the pedal with all the music I play.
In fact, I’m so used to the foot pedal for page turns, that on the few occasions when I am playing from a physical book, I’m tapping with my left foot when it’s time to the turn the page, and I get confused about why the page isn’t turning itself.
Do you just kinda hit it when the mood strikes, or do you practice hitting it at very specific times (e.g., the second beat of the last measure on the page)?
I don’t exactly “practice” hitting it any more (although I think I did when I first started using it). But one thing I have gotten into the habit of doing is making sure my left foot is in the right place. I kind of find myself moving my foot around, especially when my back hurts. So I’ll pull my left foot back away from there the pedal is because moving like that kind of gives my back a rest just by shifting position. So, depending on the complexity of the piece, I’ve found that it helps to shift my foot back closer to the pedal when I’m on the bottom half of the page, rather than trying to move my foot back into position in the last few measures of the page. I should also add that I always play my page turning pedal so that it’s sort of in a line with the piano pedals.
In my jury piece (the piece I played for an exam in May), most of the pages were fine, but there was one page where I had to make sure to move my left foot forward before I got to a specific part of the music or else I found that I would be more likely to play some wrong note. So that was something I paid attention to as I was polishing the piece.
Do you use it to read the beginning of the next page in real time, or do you memorize the first bit of the next page and just get the page changed so that it's there as you go further along?
This depends on the piece. Sometimes I write the first RH and LH notes of the next page on the bottom of the current page, sometimes I’ve memorized the first two measures of the top of the next page, it really depends. But I do generally turn early rather than late.
Using the foot pedal to turn pages is a bit of an added cognitive burden (or task overloading, as @Gombessa said) then, so is turning a physical page with your hand. But you definitely get more comfortable with the pedal over time. I’ve been using a page turning pedal for over 10 years now, but I had been playing from a tablet for a few years before that. There was a period of adjustment, especially as I worked out the best place to put the pedal, but now it’s just part of my total process.