I was taught to fly about 1/2 mile from the runway on downwind. I learned to fly in a PA12 and I used to line up the jury strut with the runway as a sight aid. I've never held a particular angle of bank, rather with eyes out the window I've used the bank angle I need to get to where I'm going. As you get the pattern tighter, the bank angles get steeper. You should fly within your comfort level, else any benefit from the tight pattern goes away if it causes an accident.
The speed of the airplane is a major factor in flying a tight pattern. What is easy in a C140, PA12, PA18, PA25 is pretty sporty with a PA28R. It's a lot easier to stay tight with something slow. I've never flown a PA38.
-- IFMU