How to split tweets into strings?

You can view string functions as complicated black boxes or you can know that arrays are simple and work from there. The things you need to program around are bounds limits (don't write 20 chars to a 16 char array) and where the terminating zero is, which you can use strlen() or check each char and check if it's zero.

char StartHere[ 142 ]; // make as a global, stuff your incoming string here 

....

// -after- filling StartHere with the twitter data, if the length is < 32 chars, pad to 32 with spaces
if ( strlen( StartHere ) < 32 )
{
    for ( byte i = strlen( StartHere); i < 32; i++ )
    {
        StartHere[ i ] = ' ';
    }
}
// you could do this or you can check lengths during print to LCD which would be more efficient
// hey, this is just example anyway --- I would use pointers but array indexes work as well.

....

// so now that we KNOW that StartHere has 32 characters even if the last few are blank spaces...

setCursor(0,0);
// lcd.println(string1);
for ( byte i = 0; i < 16; i++ )
{
  lcd.print( StartHere[ i ] );
}

setCursor(0,1);
//lcd.println(string2);
for ( byte i = 16; i < 32; i++ )
{
  lcd.print( StartHere[ i ] );
}

// mostly I just want you to see that C string arrays are nothing but arrays of chars.
// inside string functions they all deal with 1 char at a time, that's how the hardware works.