San Jose Airport 
If you are flying into San Jose, be advised the airport is actually in the town of Alajuela. If you're not planning to stay in San Jose you don't need to go there.  You can catch the bus from Alajuela or stay the night at a hostel near the airport.
The bus stop in Alajuela is called "la radial" and it's about a $3 taxi ride from the airport.  (hail a red taxi from the street and make sure it's metered and it will save you a bit of money compared to the orange airport taxis.)
If you want to stay in San Jose, head straight out of the airport and hop on a bus labelled to San Jose.  I asked locals for directions since I arrived during the day, showed a map of where my hostel was in the city and they told me when to get off the bus.  Make sure you leave the airport with some small colones (currency).  The bus drivers will not make change for large bills.
I got into the city and took a cheap taxi to my hostel.  The bus was less than a dollar and the taxi ride was just a couple dollars.  A taxi from the airport to my hostel would have been $30.

In San Jose I stayed at  Costa Rica Backpackers Hostel.  The Hostel was okay.  The beds weren't great, but the food was both yummy and decently priced and I didn't feel like exploring San Jose.

Solo traveler trip:  try to arrive to new places by day.  Get to your hostel first and explore afterwards: hostel workers are your best friends.  They will keep you safe.  If arriving by night, have a plan in place.  Most hostels offer shuttle services if you arrive late - just get in touch with them first.  Generally this is both cheaper and safer than a taxi.


San Jose > La Fortuna
There are MANY bus terminals in San Jose - all going to different cities.  Make sure you go to the correct one!  
The bus to La Fortuna leaves from the Atlántico del Norte bus station at Avenida 9 and Calle 12.  It takes about four hours the fare is about $5.  There are only a few busses per day to La Fortuna, so alternatively you could take a bus to Ciudad Quesada and transfer there to another bus to La Fortuna. It takes about the same amount of time.