(This entry is “under construction” since I have other demands on my time right now)
I encourage the question poster to consult the works of Dr. Hugh Ross of “Reasons to Believe” who is an excellent source of information of questions in this exact field.     
http://www.reasons.org/

Dr. Ross also has a number of lectures available on YouTube that covers areas exactly like this.
https://www.bing.com/videos/search?q=youtube+hugh+ross+scientific+evidence+christianity&view=detail&mid=02D614B29F37D38C366402D614B29F37D38C3664&FORM=VIRE

Here’s a few things off the top of my head
1)	 The Bible says in 
 >1 Thessalonians 5:21  21 "...test everything; hold fast what is good."<    This scripture while generally applied spiritually speaking can and has been implied for advocating a method of epistemology that is essentially “the scientific method”.

2)	There are certain Bible verses that speak of epistemology in a way that advocates some of the rules of logic.   An example

>James 3:11 Can both fresh water and salt water flow from the same spring?

This again is usually taken metaphorically or spiritually but is essentially in logic known as the “Law of Non Contradiction”

3)	Jesus in the New Testament makes a number of statements that can be used as encouraging the development of Epistemology and Empirical reason when considered literally and beyond their immediate spiritual context.    Some examples “The parable of the Foolish Man and Wise man building their houses” (Things not built on truth will have unstable foundations and not last).
Sayings about you will “know them” by their fruit (ministers and false ministers).   This verse has been employed as encouraged a certain use of empirical evaluation of things based on “the results”.
4)	A point I will develop later from Hugh Ross lectures latet, but the Genesis account and other verses of the OT speak of such things as a Singularity Beginning (Space Time Theorem), as well as other things like the Cosmic Expansion of the Universe etc.