DataGridView - how to set column width?

You can use the DataGridViewColumn.Width property to do it:

DataGridViewColumn column = dataGridView.Columns[0];
column.Width = 60;

http://msdn.microsoft.com/en-us/library/system.windows.forms.datagridviewcolumn.width.aspx

Regarding your final bullet

make width fit the text

You can experiment with the .AutoSizeMode of your DataGridViewColumn, setting it to one of these values:

None
AllCells
AllCellsExceptHeader
DisplayedCells
DisplayedCellsExceptHeader
ColumnHeader
Fill

More info on the MSDN page

i suggest to give a width to all the grid's columns like this :

DataGridViewTextBoxColumn col = new DataGridViewTextBoxColumn();
col.HeaderText = "phone"            
col.Width = 120;
col.DataPropertyName = (if you use a datasource)
thegrid.Columns.Add(col);    

and for the main(or the longest) column(let's say address) do this :

col = new DataGridViewTextBoxColumn();
col.HeaderText = "address";
col.Width = 120;

tricky part

col.MinimumWidth = 120;
col.AutoSizeMode = DataGridViewAutoSizeColumnMode.Fill;

tricky part

col.DataPropertyName = (same as above)
thegrid.Columns.Add(col);

In this way, if you stretch the form (and the grid is "dock filled" in his container) the main column, in this case the address column, takes all the space available, but it never goes less than col.MinimumWidth, so it's the only one that is resized.

I use it, when i have a grid and its last column is used for display an image (like icon detail or icon delete..) and it doesn't have the header and it has to be always the smallest one.

Use the Columns Property and set the Auto Size Mode to All Cells, Resizable to True, Frozen to False and visible to True.

The column will automatically resize based on the data inserted.

simply

dataGridView1.Columns[0].FillWeight = 20;