Central Mendoza can easily be explored on foot. For further distances, taxis are an easy, inexpensive option. You should be able to arrange a taxi for under US$10 per hour to see any sights surrounding the city.
It is possible to use the bus system to get out to the vineyards, although you should be prepared for lengthy waits and walks. You will need to get a Mendobus card, available at any kiosk in town, to ride the bus.
Renting a car is a great way to see the surrounding areas on your own schedule. Be warned that navigating can get tricky. Street signs seem to have become a prize for locals who steal them shortly after they are erected. Both Avis and Hertz have outlets in town and at the airport. Maps of the wine region are hard to come by, but here is a good one of Maipu that we found (click to enlarge):