Zakopane's town centre is walkable — Krupówki Street and the immediate surroundings can all be covered on foot. For the mountains, you will need transport.
Cable car is the main way to reach Kasprowy Wierch: the cable car from Kuźnice (a short taxi or bus ride from the centre) runs continuously during daylight hours from spring through autumn and in winter during the ski season. Queues can be long in peak season — book tickets online in advance.
Funicular to Gubałówka departs from the top of Krupówki Street. The ride takes five minutes and operates all day in season.
Minibuses (busy white vans) run frequently to Palenica Białczańska, the starting point for the hike to Morskie Oko. They depart from the bus station area throughout the day. The road beyond Palenica is closed to private cars.
Taxis are available throughout the centre and are cheap by Western European standards. For the mountains, negotiate a fixed price before departure.
On foot is perfectly reasonable for getting between the centre and Kuźnice (about 2 km), and for most of the main hiking trails.
Driving into town is fine in summer but Zakopane's roads are genuinely congested on busy weekends and holidays — the town was not designed for the traffic it now receives. Parking at the edge of town and walking in is often faster.