At the most basic level, the reminders app is like an electronic to-do list. I like to use the feature where it will remind me at a time and day to do something. I want to make note that there are additional features which I do not use. The first allows the user to set a reminder by location. The second feature allows reminders to be set on repeated schedules.
Design Problem 1: Snooze Feature
The snooze feature involves several steps so even though it is problem 1 it is also a myriad of steps and other issues. First, I want to point out a few things about the app and myself. I have a habit of setting reminders to dates and times that I often can’t realistically meet which in turn leads me to snooze them with regularity. When a reminder is due there is a visual banner on the screen and a sound (if not on silent).
One can see that it indicates to the user to “slide to view.” There is no indication which way to slide. iPhone users are familiar with sliding left to right to unlock the phone (a learned behavior – knowledge in the head), however if you slide right to left, it opens a different option to set things directly in the app without unlocking the phone and going directly to the app. This is also a learned behavior and one that does not have the best mapping. Perhaps a double arrow could be added, but this might add more complexity. A better clue for the options of operation like slide left for immediate action. I’m going to skip this issue for now and focus on snooze when sliding right to left.
If the user hits “snooze,” the reminder goes away and the lock screen image appears. The user is unaware of how long the snooze is for, one might assume similar to the alarm app on the iPhone it would be for nine minutes. It is actually for 15 minutes. The design lets the user know what actions are possible, but it does not allow the user to make use of constraints effectively. I would argue that having snooze options appear at this point would be more beneficial than appearing after the first fifteen minutes (as it does for me).
If the user hits “Later,” there are options for 15 minutes, 1 hour, tomorrow or Ignore. I would suggest letting the user adjust these snooze options by adding an additional option that would give the user more control and makes the app a little less automated. 
Once in settings, the user could then adjust the snooze to their preference (whether it be 3 minutes or 9 hours or some other increment of preference).
Design Problem 2: Order of Reminders
The reminders are by default put in the order in which they are created (not by date in which they are due) which is not always the order in which they need to be done. As a user, I find this frustrating. There is an option when setting the reminder, to give it a priority. I admit, I haven’t used this feature because of the graphic interface. The interface for me seems extreme with the exclamation points. Perhaps this need further exploration.
A possible solution to this problem would be for the user to be able to control the order in the same way the user can control apps on the iPhone. When a user presses and holds on an icon for a few seconds, it will start to jiggle and the user has the option to delete the app or move the app.
The solution for the reminders would look something like the screen shot below where the user could hold their finger on the reminder and move it around on the list by swiping it up or down once the “x” appears in the upper right corner.
This would use knowledge in the head and would allow the user some visual clues and mapping that they are familiar with to move the list into their own preferential order. Once the reminders start to wiggle, it would give the user some immediate indication of action.
Other Notes:
I wanted to add that the Reminders app does make it obvious how to add something to the list with the “+” icon at the bottom left of the screen. At its simplest, it functions as a to-do list. Once a task is completed, the empty circle is filled in with a color to indicate it is complete. This action gives immediate indication of the action’s results.