Many Southern drivers and other crew do actually drive to work, even at the South London / outer Metro depots of Selhurst, Norwood, Epsom, etc. The inner Metro depots at Victoria and London Bridge see a much more varied combination of people using staff taxis (nb. provided for certain drivers on specific duties at urban depots only!), public transport and cycling, with little or no driving into work. It is my experience that uniformed staff travelling on public transport tend to work on stations, which generally have very slightly easier shift start/finish times, which align better with the start and end of service. Bear in mind that almost all Southern train crew can and will regularly be asked to book on at any time - that includes deeply impractical times such as 0400 on Boxing Day, 0300 Sunday morning, you name it.
Once you are on duty, it is entirely the company's responsibility to provide transport within your shift. There are walking routes, these generally around stations or depots, and rarely more than 10-15 minutes in realistic terms. Each has a standard timing allowance and the same applies to taxis / replacement buses / trains / interchanges between services.
Lots of staff now use trolley cases, though the company issue ones can be carried as a backpack, and conventional backpacks can also be used. For the essentials you don't always need a bag quite that big, except if you eat a lot or like to stuff your coat in there. A crew bag can get fairly heavy and wheeling it is not a bad move. You can use your own bag, but it must not be able to snag on a moving train (no loose straps etc.) and cannot cover your orange hi-vis vest, unless it is also of the same material. You may also need to sign a disclaimer.
A lot of driver training happens at Selhurst, but you will also train with instructors based at the depot where you will eventually work full time, and there are training rooms all over the place where you may occasionally get sent - Streatham Hill, Hove, Worthing...
Go and look up the route maps and timetables, even things like Wikipedia, you name it! Speak to some staff or go travelling around a bit on the network. However much bad press Southern gets, they have some cracking routes and stations, and most staff are perfectly friendly if you find a quieter moment to have a chat.