Question: Is there any way to show a calculated field when I'm filling out a new item for a list?

For example:

If I select "Blue" in field1, and "Bird" in field2,

then, on the same page where I am filling in information, I can see field3(Calculated field) show a value of "Blue Jay"

Currently, the calculated field doesn't show until I add a new item.
It would be even better if the update was in real time, though I don't expect that.

Answer: 
**As a short answer** : unfortunately , No, the calculated field is calculated after the item added or updated 

If you are using Enterprise Edition of SharePoint then try editing list form with InfoPath and insert field which will do the calculation for you. Make that field read-only and then publish the form.

In InfoPath , You can 

* Add a Textbox as `field3` in your form
* At `field2` add new a `Action rule` and at **run these action select** `set a fields value`
* At **Field** Value select Field3
* At **Value** concatenate your `field1` and `field2`

[![enter image description here](https://i.stack.imgur.com/SN6jP.gif)](https://i.stack.imgur.com/SN6jP.gif)
Calculated columns don't work that way, they are visible on the display form only or in views and only recalculate when items are edited.

If you want that type of preview feature, you'll have to incorporate custom javascript on your forms.