The different regiments of Lawton's Brigade took trains from Savannah Ga to Charleston SC. From Charleston the rail line led to Florence SC then to Wilmington NC. Troops would have boarded another train at Wilmington as the rail guage north differered from the rail gauge from Florence. From Wilmington the railroad took the troops to Goldsboro NC, then to Weldon NC. From Weldon it was a relatively short distance to Petersburg VA.

Here is the order directing Lawton's Georgia Brigade west to Lynchburg VA:

RICHMOND, VA., June 10, 1862.

Brigadier-General RANSOM, Petersburg, Va.:
The brigade under General Lawton is destined for General Jacksons army, near Port Republic. As the several regiments arrive at Petersburg cause them to be sent by railroad via Lynchburg to Mechums River Depot.

S. COOPER, Adjutant and Inspector General.

General Lee wrote Jackson on June 11th that Lawton's Georgia Brigade had been sent to his army.
See Official Records, Series 1, Vol 11, pages 585 and 589.

From Lynchburg the troops would have traveled to the rail junction at Charlottesville VA.
http://en.wikipedia.org/wiki/Southside_Railroad_(Virginia)
From there the line led west to Staunton, and east to Gordonsville VA. At this point Lawton's Briade became part of General Jackson's command on its route to join Lee at Richmond.