Tenerife North airport is next door to the city, a short hop by bus from La Laguna Bus Station.
The town is fairly easy to walk around. There is now a tram running from downtown Santa Cruz direct to La Laguna (Line #1). The closest stop to the old town is the terminus of the route (La Trinidad). It takes about 30-40 minutes and costs about €2.35 return. A 14 or 15 bus will get you there from Santa Cruz and many buses stop at the bus station. This is maybe ten minutes walk down to the town itself or get a 14 or 15 bus and get off at the first stop.
Some excellent cake shops at great prices.
Since there are so many students in the area, almost no accommodation available for tourists. The central Hotel Nivaria offers apartment rooms with kitchens and internet access (bring an ethernet cable).
The new tram system takes you to Santa Cruz bus station in about 40 minutes for €2.35 return. There are also frequent buses taking you from the new bus station close to Avenida Trinidad: number 015 directly to Santa Cruz via highway in just 15 minutes or 014 which takes more time as it goes through the city.