My laserdisc player, a Pioneer CLD-59 Elite player occasionally gives me a U1 warning and spits the disc out at me. Today it did it twice though, which worries me. It U1'd me when it turned the player on and went straight to the B side, and it did it again just now, as I closed the tray by turning the player off. Can anyone help an electronics amateur like myself solve this issue, or am I worrying too much.... Getting that error during shutdown really wigged me out....

You can get this if you have not inserted the disc centrally in the tray or during side change if there is a misalignment issue rotating the laser pickup to play side B.
Other than that I would not expect to see that error very often. If it happens regularly then you should get your player checked out.

It only happens once every 15 times I use the player

A U1 error means there was a misclamp. A U1 error in my experience is a belt error. It can be a lot of things, but if you are getting it intermittently its probably just the belt. You can usually make this error occur by trying to play heavier LDs. DTS discs are a little bit heavier and made the occur more on my player. The best disc at getting the error to occur was The World on a Silver Platter, a sales training LD. One side was hard white plastic so it was the heaviest disc in my collection.

Anyway the solution to this problem most likely is to remove and clean the loading belt, or to outright replace it. This is one of the easiest things you can fix on an LD player. It only involves removing the catches on the tray, pulling the tray out all the way, then replacing the belt. Every LD player is a little different in how you remove the tray. Correct me if I'm wrong but isn't a CLD-59 very similar to a d704?

The CLD-59 is similar to the CLD-D604 but yes it has the same belt loading issue as you described with the D704.

I agree, take off the top, open the tray and then pull out a little more and you'll start to see the belt on the front left just under the tray. Clean or replace the belt and clean the pulley on the loading motor.