I have 3 columns in my DataGridView. What I am trying to do is have the first 2 columns auto fit to the width of the content, and have the 3rd column fill the r         
        
This is my favorite approach...
_dataGrid.DataBindingComplete += (o, _) =>
    {
        var dataGridView = o as DataGridView;
        if (dataGridView != null)
        {
           dataGridView.AutoSizeColumnsMode = DataGridViewAutoSizeColumnsMode.AllCells;
           dataGridView.Columns[dataGridView.ColumnCount-1].AutoSizeMode = DataGridViewAutoSizeColumnMode.Fill;
        }
    };
This is what I have done in order to get the column "first_name" fill the space when all the columns cannot do it.
When the grid go to small the column "first_name" gets almost invisible (very thin) so I can set the DataGridViewAutoSizeColumnMode to AllCells as the others visible columns. For performance issues it´s important to set them to None before data binding it and set back to AllCell in the DataBindingComplete event handler of the grid. Hope it helps!
private void dataGridView1_Resize(object sender, EventArgs e)
    {
        int ColumnsWidth = 0;
        foreach(DataGridViewColumn col in dataGridView1.Columns)
        {
            if (col.Visible) ColumnsWidth += col.Width;
        }
        if (ColumnsWidth <dataGridView1.Width)
        {
            dataGridView1.Columns["first_name"].AutoSizeMode = DataGridViewAutoSizeColumnMode.Fill;
        }
        else if (dataGridView1.Columns["first_name"].Width < 10) dataGridView1.Columns["first_name"].AutoSizeMode = DataGridViewAutoSizeColumnMode.AllCells;
    }
public void setHeight(DataGridView src)
{
    src.Height= src.ColumnHeadersVisible ? src.ColumnHeadersHeight : 0 +   src.Rows.OfType<DataGridViewRow>().Where(row => row.Visible).Sum(row => row.Height);
}
Try this :
  DGV.AutoResizeColumns();
  DGV.AutoSizeColumnsMode=DataGridViewAutoSizeColumnsMode.AllCells;
To build on AlfredBr's answer, if you hid some of your columns, you can use the following to auto-size all columns and then just have the last visible column fill the empty space:
myDgv.AutoSizeColumnsMode = DataGridViewAutoSizeColumnsMode.AllCells;
myDgv.Columns.GetLastColumn(DataGridViewElementStates.Visible, DataGridViewElementStates.None).AutoSizeMode = 
    DataGridViewAutoSizeColumnMode.Fill;
Not tested but you can give a try. Tested and working. I hope you can play with AutoSizeMode of DataGridViewColum to achieve what you need.
Try setting
dataGridView1.DataSource = yourdatasource;<--set datasource before you set AutoSizeMode
//Set the following properties after setting datasource
dataGridView1.Columns[0].AutoSizeMode = DataGridViewAutoSizeColumnMode.AllCells;
dataGridView1.Columns[1].AutoSizeMode = DataGridViewAutoSizeColumnMode.AllCells;
dataGridView1.Columns[2].AutoSizeMode = DataGridViewAutoSizeColumnMode.Fill;
This should work