I think this line :
var sunsettimes = [10/1,"7:19",10/2,"7:17",10/3,"7:15"];
is the main problem as the dates you put in are interpreted as operations (for the first one, 10 divided by 1.
Changin the line to:
var sunsettimes = ['10/1',"7:19",'10/2',"7:17",'10/3',"7:15"];
So the dates are used as Strings should do the trick.
Also, as you said you're new to this, I'll allow myself to give you some advice/knowledge. The system you use (a list with both the keys and the values side by side) seems weird as there are already some things taking of this problem perfectly in Javascript. Those things are the Objects.
This is how you would use them:
$(function() {
var date = new Date();
//need to use a date object
var month = date.getMonth() + 1;
var day = date.getDate();
var todaysdate = month + '/' + day;
var sunsettimes = {
'10/1': "7:19",
'10/2': "7:17",
'10/3': "7:15"
};
//need to get the value at index indexsunset
var displaysunsettime = sunsettimes[todaysdate];
$("#sunsetbox span").html(displaysunsettime);
});
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>
<div id="sunsetbox">Sunset tonight at <span></span>
</div>
The objects allow you associate a value to a key, and they will return the value associated to a key you pass them.
I hope i was clear enough
daythere is no/in the array valuegetMonth()on their own.