Bark chips are a good choice of mulch for under trees and shrubs and a depth of 3-4 inches (ideal, 2" minimum) It will retain moisture which will reduce the need for watering and also supress weed growth mainy by blocking out the light.
If you have previously cleared the ground of weeds the mulch should last for 2-3 years and only a little hand weeding required.
If you do decide to use polythene sheeting, make sure it's a semi permeable membrane ie woven polypropolene, which will block out light, allow water to perculate through but stop weeds coming through.
Seeing that you already have established trees and shrubs, I'd go for the no membrane option but if you do, make sure that the mulch is not touching the base of the shrubs otherwise it can cause them to rot, just leave a small area clear around the stems. Good luck. Tbird+