I am trying to figure out what types of things are best tested in my website(s), phone apps, or programs. These need to be things that can actually be measured. Which information is most useful to UX/UI designers?
I assume that by statistics you mean analytics.
There are a lot of interesting variables that I use when starting work on new projects.
Basic stuff like which browsers are in use, what screen resolutions the target groups use what times of day they visit, how often they visit all tie in with whatever qualitative research you may be doing to help form a more complete picture of who you are designing your product for.
Also a lot of analytics suites are incorporating pathing-tools which allow you to see the paths users take around your app. This is super helpful in settling disputes about the relative importance of pages and it can help eliminating a lot of the hypotheticals.
All these things help you answer what people do, but not really why they do it.
Presuming that you're talking about analytics or some other method of measuring what is actually clicked on, you should work with your UX designers and researchers to determine what they need measured. They might have questions about how frequently something is used, whether something is accessed more frequently in one way over another, the order in which various things are used, and many more.
If you don't have UX designers/researchers and are just hoping to do this yourself (that is, imagining what hypothetical UX designers/researchers would do with this data), then this should be a collaborative effort between you and other stakeholders in the project. If you can't get the stakeholders to agree, then find other people on the team who are interested in this and brainstorm with them some items that would be beneficial for you to measure.
When deciding on what you should measure, the question that you should keep in mind is "what will we do with the data that we gather?" For example, if you decide to measure how frequently a particular feature is used, what will you do if you find out that it's used infrequently? Does infrequent usage mean that this feature is difficult to use, is difficult to find, or is something that is not done frequently but is very important when you do need to do it? In each of those cases, you will have a different solution: make the feature easier to use, make the feature easier to find, or leave it alone because (even though it's infrequently used) it's meeting users' needs.