How to add a string to a string[] array? There's no .Add function

You can't add items to an array, since it has fixed length. What you're looking for is a List<string>, which can later be turned to an array using list.ToArray(), e.g.

List<string> list = new List<string>();
list.Add("Hi");
String[] str = list.ToArray();

Alternatively, you can resize the array.

Array.Resize(ref array, array.Length + 1);
array[array.Length - 1] = "new string";

Use List<T> from System.Collections.Generic

List<string> myCollection = new List<string>();

…

myCollection.Add(aString);

Or, shorthand (using collection initialiser):

List<string> myCollection = new List<string> {aString, bString}

If you really want an array at the end, use

myCollection.ToArray();

You might be better off abstracting to an interface, such as IEnumerable, then just returning the collection.

Edit: If you must use an array, you can preallocate it to the right size (i.e. the number of FileInfo you have). Then, in the foreach loop, maintain a counter for the array index you need to update next.

private string[] ColeccionDeCortes(string Path)
{
    DirectoryInfo X = new DirectoryInfo(Path);
    FileInfo[] listaDeArchivos = X.GetFiles();
    string[] Coleccion = new string[listaDeArchivos.Length];
    int i = 0;

    foreach (FileInfo FI in listaDeArchivos)
    {
        Coleccion[i++] = FI.Name;
        //Add the FI.Name to the Coleccion[] array, 
    }

    return Coleccion;
}