How to put commands for minimum, maximum and average of numbers in different methods

This would work as a way to encapsulate your finding the minimum as a separate method. (And similarly for maximum and average.)
Your original code is more efficient than using separate methods, (even if you had loading the values into the array as a separate loop). It iterates over the elements of the array only once (twice), instead of three (four) times.

Thanks for the advice Matt, however, one of the requirements of this assignments is to use different methods to perform each of the calculations. I tried the following code but cannot figure out the error. Could you identify the problem?:

public static int find_min(int[] a)
{
int min = int.MaxValue;
for (int n = 0; n < a.Length; ++n)
{
if (min > a[n]) min = a[n];
}
return min;
}

public static int find_max(int[] a)
{
int max = int.MinValue;
for (int n = 0; n < a.Length; ++n)
{
if (max < a[n]) max = a[n];
}
return max;
}

public static double find_sum(int[] a)
{
sum = 0;
double avg = (double)sum / NUMBERS;
for (int n = 0; n < a.Length; ++n)
{
sum += a[n];
}
return avg;
}

Your min and max look correct.
Your find_sum (which looks like it is really supposed to be find_avg!) has a problem.
The statement double avg = ... should be after the loop where you accumulate the sum, otherwise you always return 0.0
Also, the method should use a.Length instead of NUMBERS for the divisor!
(And you need to declare the sum local variable:
int sum = 0;
)

Thanks for the corrections, much appreciated. However, I still keep getting the error : "The name 'sum' does not exist in the current context."

I updated my reply above:
And you need to declare the sum local variable:
int sum = 0;

As a result of adding the different methods, could anything be omitted from the original solution?

Of course!
The while loop should only set the values into the a array.
Then use the three methods you now have, after the loop, to get the values of the min, max, and avg.
The sum variable in the main is now useless and should be removed.

So would the while loop now look like this?:
while (i < NUMBERS)
{
a[i] = myRandom.Next(1, 501);
++i;
}

Yes

I tried the following code but it says that 'avg'does not exist in the current context, when trying to print "The average of these numbers is."

using System;
using System.Linq;

public class Test
{
public static void Main()
{
const int NUMBERS = 1000;
Random myRandom = new Random();
int [] a = new int[NUMBERS];

int i = 0;
int max = int.MinValue, min = int.MaxValue;

while (i < NUMBERS)
{
a[i] = myRandom.Next(1, 501);
++i;
}

Console.WriteLine("The minimum number is: {0}", min );
Console.WriteLine("The maximum number is: {0}", max);
Console.WriteLine("The average of these numbers is: {0}", avg);

int UserInput = Convert.ToInt32(Console.ReadLine());

if(a.Contains(UserInput))
{
Console.WriteLine("Your number was one of the original 1000!");
}
else
{
Console.WriteLine("Your number was not one of the original 1000");
}

}
public static int find_min(int[] a)
{
int min = int.MaxValue;
for (int n = 0; n < a.Length; ++n)
{
if (min > a[n]) min = a[n];
}
return min;
}

public static int find_max(int[] a)
{
int max = int.MinValue;
for (int n = 0; n < a.Length; ++n)
{
if (max < a[n]) max = a[n];
}
return max;
}

public static double find_avg(int[] a)
{
int sum = 0;

for (int n = 0; n < a.Length; ++n)
{
sum += a[n];
}
double avg = (double)sum / a.Length;
return avg;
}

}

I told you, above, that you have to use the methods to get the values of the min, max and avg. That means you'll have to declare the variables, of the correct type, to store the results of calling those methods.
This is just basic programming!
You seem to understand how to use methods to get the random values and display results to the console. Call your methods similarly.

Thanks, that worked perfectly!