How do I make the refresh happen AFTER I click on the button?

One approach: You can save the value of the search field into a local variable and the value of this local variable will get updated when the field becomes unfocused. Then you can have some logic in the saveInto of the button to filter the data using the local variable that holds the search criteria.

@erickp The ideal way to achieve the use case would be to shift the logic of querying the database with the given filters,constructing a datasubset and paginating thereby to the 'saveInto' of the button(the button that will be clicked after providing values to the filters). Probably this logic would have resided in the with() or 'saveInto' of the search fields as of now.

@erickp: The issue you are facing here is not about page refresh. Its about where you have put the code to populate the Grid. Currently the grid is getting populated on refresh of Search field. You need to move the code such a way that it gets executed on click event of the button.
So as Joshi and Sikhivahans mentioned, use the 'saveInto' of the Button to populate the Grid.

Not sure if I understood. Where do I need to move the code so that it gets executed when clicking the button?

Attach your code here. Will try to help you by modifying the code.

@aviroop I am attaching the code. To populate the grid, I am using a query rule and storing the variable in local!universities. Notice that I have that inside a load. That is another problem I am having. I know it has to be on a with() so that when I do a search, the grid updates. However, notice that I have a separate rule (attached text file called row) inside the applyComponents() to generate the rows. In that other rule, I get an error if I put my local!universities variable inside a with(). It tells me that the save function can't accept a variable coming from a with(). Please advice what is best to fix the problems

Try this code out. When you click on the "Apply" button, grid should get populated. (I didnt find any button labeled "Search" in your code. So implemented the functionality on "Apply" button click)

main_rule.txt

Excellent. That makes sense. It worked!

It does work when I test it out on the designer. However, I publish my application, it doesn't work once I try doing a search. I am trying to figure out what should I change in my process flow so that once they type a university name and click Apply, the university results change. I am missing that part.

Its very hard to tell since I dont have any insight of your process model and any other rules which are used to pull data. If its working properly on designer, then the logic is fine.
You need to check the node's i/o variables, pv and other rules to get it fixed in Application flow.