The only ways to get to Aguas Calientes are by train or by foot.
On foot
If you're on a budget, or just adventurous, it's possible to hike downstream along the railway tracks from Ollantaytambo or from the town at km 82, where the Inca trail starts, this is about a seven hour hike Note - hiking on the train tracks is prohibited.
It's also possible to hike upstream along the train tracks from Santa Teresa 4 hours or Hydroelectrica 2 hours. You can take a minibus directly to Hydroelectrica from Cusco for S/50 or make the journey by public transport: To reach Santa Teresa, take a bus towards Quillabamba from Cusco and get off at Santa Maria. The bus leaves Cusco at 8am from the Santiago bus depot - S/20 and passes through Ollantaytambo, Urubumba and Santa Maria. It´s an 7 hour journey from Cusco to Santa Maria. You can also take a Minivan Colectivo which leaves next to the buses S/25-30 - 4 hours. At Santa Maria, take a connecting bus to Santa Teresa S/6, 1.5 hours or a taxi S/10, 1 hour. Walk 2 hours or catch a bus S/5 to the hydro electric plant planta hidroeléctrica. From there it's two hours of walking to Aguas Calientes from here, but it's also possible to catch a train for $18.00 US to Machu Picchu Pueblo, possibly much cheaper if you are Peruvian leaves hidroeléctrica at 4 PM. As the tracks are still in use, be careful, especially when crossing bridges and in the tunnels.
There are also hiking paths coming from Mollepata, Cachora and Huanicapa for the extremely adventurous. You will want to get your hands on some topographical maps beforehand, Hiking and Treking around Cusco is available for around S/.25.00 to S/.30.00 and has details on the routes you can take.