As far as I know there is only a bus to the Volcán Irazú during the weekends. This bus goes from San José and from Cartago then. But: when we visited the volcano it was a weekday so the bus was no option.
Instead, we decided to take a taxi to the top of the volcano. After a short discussion with a taxidriver in Cartago we agreed on a price of 20 dollars for the two of us for a return trip plus one and a half hours free time on the top.
The taxi takes you through the entrance gate of the park, where you have to pay for your tickets, and then you drive to the parking lot at only 300 meters from the main crater of the Irazú.
Cartago is located just east from the capital of Costa Rica, San José. It takes only a short busride to get there: in less then 30 minutes (depending on the traffic in the capital) you can be in Cartago already. And busses depart every day, every hour, all day long.
The busses to Cartago depart at the corner of Calle Central and Avenida 3 and cost you only 50 cents. Tickets can be bought in the bus itself, so you don't have to look for any ticket office.
When you want to go back to San José, take the bus in Cartago just north from the Parque Central.
A very nice trip from Cartago is a visit to the beautiful Orosí Valley. It can be done pretty easily because the trip by bus only takes 40 minutes (and takes you through a great landcape too).
The best way to get to the Orosí Valley is by taking the bus that departs just south-west of the Church Ruins in Cartago. It is a flashy green bus, with line 339 on it. You can pay for your ticket in the bus: it costs about 1 dollar to get there.
When you want to get back, take the bus at one of the few busstops in the centre of the village. Busses in Costa Rica are not the same as in the rest of Central America: here they don't stop just anywhere when they see passengers, they only stop at the busstops.
If you plan on vistiting the Irazu Volcano rent a car to get there. The hotels in town charge $70-90 dollars per person for a tour but by all means don't do this. If you rent a car it will be about $55 and its an adventure driving in Costa Rica just remember you have to be patient with the local drivers.