Mt Bromo is the most favorite Travel destination in Indonesia as one of the most iconic mountain in Indonesia. 
Most travelers visit Bromo from the base town of Probolinggo. The access to Probolinggo its relatively easy and proximity to airports in Surabaya and Malang.   Book Bromo Tours

Unfortunately, travelling from Yogyakarta to Mt Bromo is more difficult.
There is no cheap and easy way from Yogyakarta to Mt Bromo, so here are some suggestions to maximise comfort or minimise cost.

Option

#1. Backpackers Style
BUS TO PROBOLINGGO - PUBLIC TRANSPORT TO BROMO
Cheap? YES  ,  Comfortable? NO

Buses from Yogyakarta to Denpasar (Bali) will usually stop in Probolinggo, but check first.

The trip would be 10-12 hours and the Giwangan bus terminal is 5km from central Yogyakarta, so again it would be an early start to ensure you arrive in Probolinggo by mid-afternoon then continue with using public transport to bromo 

Bus tickets cannot be booked in advance and please check the bus schedule at the bus terminal


#2. Traveler Style
TRAIN TO SURABAYA - BUS TO PROBOLINGGO - PRIVATE / PUBLIC TRANSPORT TO BROMO
Cheap? YES  ,  Comfortable? YES & NO

The most popular choice is to take the morning train to Surabaya from Yogyakarta, then travel by bus or taxi to the Bungarasih bus terminal (near the airport) and get a bus to Probolinggo.

Please note that direct buses to Bungarasih rarely travel past Gubeng Station in Surabaya; if you are keen, you would exit the station from the west/left side of the train. Alternately, you can take a taxi a short distance to Jl Panglima Sudirman (Jl = Jalan, Street); there are more buses to Bungarasih  (including air conditioned coaches) passing by here.

Taxis are available on the east/right side of the train; the most reliable are Bluebird taxis, waiting across the road from the car park


#3. Overland Tour
PRIVATE TRANSPORT / GROUP TOUR TO MT. BROMO
Cheap? YES & NO  , Comfortable? YES & NO

Many travel agent offer this package. Usually you start at early morning from Yogyakarta and you will arrive in Bromo by mid-afternoon

The trip would be takes 10-12 hours.


#4. Premium
FLIGHT / TRAIN TO SURABAYA or MALANG - TOUR TO MT BROMO
Cheap? NO   ,  Comfortable? YES

Prices for tours vary significantly according to the number of passengers, level of comfort and standard of hotel.

The journey from Surabaya to Bromo takes about 3,5 hours driving.

Book Bromo Tours

Source : Internet review & Tripadvisor's forum