Hit Control-N to create a new document. Select Pixels from the Units drop-down menu, enter 600 in the width and height boxes and then click the Advanced button. Select RGB, Screen (72ppi) and make sure that the Align New Objects to Pixel Grid box is unchecked before you click OK.
Enable the Grid (View > Show Grid) and Snap to Grid (View > Snap to Grid). You will need a grid every 5 px, so simply go to Edit > Preferences > Guides > Grid, and enter 5 in the Gridline every box and 1 in the Subdivisions box. Try not to get discouraged by all that grid—it will make your work easier, and keep in mind that you can easily enable or disable it using the Control-" keyboard shortcut.
You can learn more about Illustrator's grid system and how it can make your work easier in this short tutorial from Andrei Stefan: Understanding Adobe Illustrator's Grid System.
You should also open the Info panel (Window > Info) for a live preview with the size and position of your shapes. Don't forget to set the unit of measurement to pixels from Edit > Preferences > Units > General. All these options will significantly increase your work speed.
If you're looking for a different type of design, you can find a variety of logo designs and templates on Envato Market.