Quweira sits on the Desert Highway (Route 15), the main north-south artery running between Amman and Aqaba. It is about 50 kilometres north of Aqaba and roughly 290 kilometres south of Amman. Any bus or minibus travelling this route will pass through or close to Quweira, though not all will stop in town.
If you are heading to Wadi Rum, the turnoff leaves the Desert Highway roughly five kilometres south of Quweira, at a signed junction at around 29.725°N, 35.283°E — about 40 kilometres north of Aqaba. From there, the road runs east for about 21 kilometres across open desert to the Wadi Rum Visitor Centre. There is no public transport along this road. Hitching is possible but not guaranteed; tour operators in Aqaba sometimes run transfers that pass through the junction. The most reliable option is to arrange a pick-up in advance with your camp or guide in Wadi Rum.
If you are coming from Aqaba, shared taxis (service taxis) run north along the Desert Highway and can drop you at the Wadi Rum junction. From there you will need to arrange onward transport or walk the 21 kilometres, which no one does.