There are a couple of songs I suggest playing for new Hard players...
When You Were Young...
Bulls on Parade...
Some of the most fun songs I suggest playing:
My Name is Jonas (TIP: On Hard in the beginning, you only need to strum first note and press the note frets until the double notes.)
Suck My Kiss (One of the easiest and most fun!)
Don't Hold back (BONUS)
3's and 7's
Generation Rock (BONUS)
Prayer of the Refugee (BONUS)
Cult of Personality (Sweet Solo)
Through the Fire and Flames (Holy ****!)
On notes that seem to be "shinier" or brighter you only need to press the fret buttons down, but it seems you need get a combo started by strumming at least one note before only pressing the fret buttons.
^^TIP^^ If you are a beginner don't focus on this, and if there are many strummed notes with only one hammer-on or pull-off don't try to do it unless you are experienced.
In battles if you get Double Notes, or Difficulty Up, use it right before your opponents notes come on the screen.
Try going from Easy-Medium/Medium-Hard/Hard-Expert/Expert-Hyperspeed/ Hyperspeed(1-5) the harder it gets the more fun it is...
I hope I helped some beginners on Guitar Hero III!