Getting out of Laayoune town for a bit of sightseeing did not prove very difficult, but for sure it was expensive. because of the lack of tourists, there's no competition - so only one hotel (the Parador) offers taylor-made day tours, which means that you get a (pink) 4x4 and a driver - not very helpful or friendly, in our case. We paid around 120 euros for one day.
There are really no alternatives: small taxis can't really go further than the airport, and the few big taxis work as shared taxis and travel only long distances on fixed routes. You can contact the hotel Parador through any hotel in town, or yourself by phoning the following +212 48 892814
Address: Rue Okba Ibn Nafia
Unless you are heading into mauritania, maybe it's worth considering flying into western sahara... sometimes, if you book well in advance, royal air maroc has affordable tickets, there. I think we paid about 80 euros for our return trip from casablanca.
However, we found that the airline is not really to be trusted too much - so do yourself a favour and call them up a few days in advance and seee if their flight is confirmed. We had a ticket to dakhla and once we arrived at the airport we discovered that the flight had been cancelled over a month earlier and that they had just forgotten to inform us... Getting them to put us on another flight was a complicated story.... hours of arguments of discussions. Eventually we managed to have them put us on a plane to a different western sahara destination - but it wasn't an easy task. It also involved calling the airport police.