Driving yourself
Driving yourself is not really recommended for the western tourists. There are hardly any rules and signs and the Peruvian drivers drive really aggressive, including a lot of hooting. Additional fact is that the roads aren't as good as in the more developed countries. Only when having a lot of driving experience and being really self assured you can try it.
Taxis
In all places taxis are to be found. Never agree with the first mentioned price by the taxi driver. At night, certainly for a woman by herself, take an official (that means it is registered) taxi, for the non-official can be dangerous. The official taxis can be recognized by the green Taxi-sticker on the front window. In most cities when being in a hostel, the hostel will probably help you with calling a taxi if that is needed. 
Minibuses
In the main cities minibuses and buses a bit larger drive to transport locals from place to place within the city. Of course also tourists can use these buses but it is not always very clear what the buses direction is (certainly when not understanding Spanish). Also minibuses drive from town to town but they often wait until the bus is full, and than we mean really full! This transportation is the cheapest but also the less comfortable.
Large distance buses
Most main cities in Peru have a bus terminal (terminal de bus) from where you can buy trips to other larger cities. This is possible by local transportation and more luxurious service. Local transportation is clearly cheaper but also much less comfortable. The more luxurious services are also not very expensive but a lot more comfortable.
Train
In Peru there are two main train routes for tourists. The most used is the service that links Arequipa, Puno and Cusco (and the part from Cusco to Machu Picchu). For prices, and more look at www.perurail.com.
The other service runs from Lima to Huancayo and Huancavelica.