I also primer every piece that is not chrome or clear plastic. I generally primer and paint everything on the sprue except for the engine. Once you remove the piece from the sprue its easy to remove the remnant of the sprue with a sanding stick for filing nails and then touching it up with a brush.
I use alligator clips for the more substantial pieces at times and they generally won't damage the piece if you don't snap them on but clamp the clip slowly and carefully. I also use tape and index cards. Simply loop a piece of masking tape, apply it to a card and them hold the card with a clip securely taped to a piece of sprue. Great for interior pieces like inner door panels and seats or even an engine piece like an oil filter. Simply attach the oil filter to the tape for the end that will be glued to engine. Then you can paint it by rotating the index card. I prefer to paint everything with a spray lacquer, outside of course. I use WalMart ColorPlace gray primer which is inexpensive and does a great job. As with all spray paint, doing it outside generally provides enough ventilation and is much cheaper than investing in a ventilated paint booth but requires good weather.
I usually assemble the basic engine pieces before paint; engine block, oil pain, heads, water pump, and valve covers, if not chromed. Then I can insert a square to round toothpick in the end of the transmission and put some tape around the toothpick and the end of the transmission. The transmission will generally be painted steel once the engine is painted and can be done later with a brush. But once the engine black is together both primer and paint can be applied while rotating the engine on the toothpick.
Hope that helps!