Is it better to take lots of little naps or to try to stay up to get on new time zone when travelling far?
No data. No clear answer to your question. My guess is staying up and not napping is more likely to get you back on track quicker, but if necessary a nap probably won't hurt.
Stay up. Jet lag is a mismatch between your internal clock and the external world if possible begin to shift to the new time zone (or at least a few hours closer to new zone) prior to leaving. Once you depart immediately begin eating and sleeping according to new time zone. Once you arrive get out. Into the morning and afternoon sunlight and take melatonin at bedtime, these help to reset your clock.