When can I request flight following? I'm a student pilot doing all VFR flying and making flight plans for training. I've heard that most people don't file a flight plan for VFR outside of training though, and instead request "flight following" from nearby ATC.

When can I do this in flight? Which ATC can do this for me?
As SkipMiller said, you can basically request flight following at any time and any certificate level - student, sport pilot, private pilot - the service is available to anyone.

Also:

We're told that FF is "workload permitting".
I used to think that meant they'd give you services when they can.   But I was wrong about that. Instead, once they accept you, you're in the system, and get full services.   If they get too busy, they'll tell you and kick you out of the system.  ("Radar Services Terminated; Squawk VFR")

FF depends on radar services.   When flying rather low and near mountains, its easy for them to lose you.  

As for when you should request it:  You can request it at anytime.
Personally, I wouldn't be in a hurry to request it as soon as you depart.   Wait until you're established on course, in a cruise-attitude (instead of a climb), and everything is settled and running smoothly.  Then know your position, and how you're going to report it.  Know where you're going (both by name and ICAO identifier), and try to anticipate any questions they might ask.

Look for the Center/Approach/Depature where you are  (look in your AF/D for an airport close to you), and call up with:

You:   "Seattle Center; November 12345: VFR, with request"
translation:  Calling Seattle Center, this is N12345.  I'm VFR (so you haven't heard from me before... don't be surprised!), and I have a request.

Center:  "November 12345: Say request."

You:  "November 12345 is SkyHawk, 10 mi. South of Harvey Field, 4,500 feet.  Request VFR Flight Following to Friday Harbor"

translation: I'm a Cessna Skyhawk, my location and altitude will help you find me.  I'm asking for FF to this other airport."

Center:  "N345: Squawk 4532 and Ident."

You: (Set your transponder to 4532, press the Identify Button), and say "4532 and Ident, N345"

Center:  "N345: Radar Contact, 11 mi. south of Sierra Four Tree, at 4,500.  Traffic at your 1o'clock, 5 mi. North Bound, Altitude indicates 3,000."

translation:  I found you on my radar scope, and am describing the position I see you. (If you disagree, speak up!).  You have traffic ahead and slightly to your right.  Their transponder indicates 3,000 feet, but I cannot independently verify that.