How can I make it so that after I enter numbers in a text input box and press the execute button, it displays the sum of each number in the text box below?
How many textboxes do you plan to use?
For only a few (in the example 3), just do it like this in the button click event:
Set Textbox4.Text to Textbox1.Text + Textbox2.Text + Textbox3.Text
A very good way to learn App Inventor is to read the free Inventor's Manual here in the AI2 free online eBook App Inventor 2 Book: Create Your Own Android Apps ... the links are at the bottom of the Web page. The book 'teaches' users how to program with AI2 blocks.
There is a free programming course here Course In A Box and the aia files for the projects in the book are here: App Inventor 2 Book: Create Your Own Android Apps
How to do a lot of basic things with App Inventor are described here: How do you...? .
Also do the tutorials Our Tutorials! to learn the basics of App Inventor, then try something and follow the Top 5 Tips: How to learn App Inventor
Taifun
Trying to push the limits! Snippets, Tutorials and Extensions from Pura Vida Apps by 
Taifun.
If you can enter them all together separated by \n (new line characters), you can sum the numbers in the button click event.
Split the textbox text at \n to get a list of numbers.
Use the sum of list block to get the sum.
Display it.