Bus nr. 34 goes from ZaragoZa Delicias train/bus station to cental market (near Basilica de Nuestra Senora del Pilar) and on to Paseo Maria Agustin/Av, Cesar Augusto (for Aiport bus).
Bus nr. 51 goes from ZaragoZa Delicias train/bus station to Palacio de la Aljaferia and on to Paseo Maria Agustin/Av, Cesar Augusto.
I went to Zaragoza a couple of random weekends, including the Fiesta del Pilar.
I came from Tarragona by car. Finding parking on the other visits was no problem. However, even during the Festival weekend, parking was not impossible. Just allow yourself a bit extra time to drive around to find a proper place and a few extra minutes to walk to the center. I think it only added ~45 minutes to find parking and walk to the center. But having the flexibility to come/go as I wanted was nice compared to waiting at train/bus station.
There is a large underground parking garage near the Plaza del Pilar.
You can reach Zaragoza by plane, train or car. Ryanair put Zaragoza 'in the map' for some inernational visitors, and definitley my friends and relatives living there are very happy the possibilities they have now to easily visit some European destinations.
However, most visitors may arrive to Spain at Madrid, or maybe Barcelona, so, if you are coming from one of these cities, you may reach Zaragoza by train (have a look at www.renfe.es) or by bus ( www.alsa.es) Even fast trains are really good and comfortable, buses are much cheaper and pretty good too (at least for a 3 hours trip). However, the landscape from the bus is pretty boring (all highways). The trip by train is more scenic.
You can reach Zaragoza by plane, train or car. I recomend train or car if you are coming from Madrid or Barcelona. You will found good and fast trains and a modern highway from these cities.
In Zaragoza the best transport are your feet and the bus, but if you want, you have taxi wherever you go