Garuda Indonesia : domestic and international destination. Route map, see here. *my favorite airlines
Lion Air : mostly domestic and some international destination. Route map, see here. Wings air is subsidiary of Lion Air for some route. *my favorite airlines
Sriwijaya Air : domestic destination.
susi air : some domestic destination. see the scheduled flight here. my experience : i see this airlines, while i was in Tarakan, Kalimantan. note : book by phone / buy in the airport.
Merpati Air : mostly domestic (if your destination to East of Indonesia, maybe you can see this airlines first). Route map, see here. my experience : i use this airlines when i went from Bali to Labuan Bajo.
Trans nusa : domestic destination. Java, Bali, Lombok, sulawesi, alor, flores, sumba, sumbawa. Flight schedule see here. note : book by phone / buy in the airport.
Air Asia : domestic and international destination. budget traveler, maybe you want to check this airlines. *my favorite airlines
Train is active transportation only in Java and Sumatera (so far i know, south and north sumatera only). Train schedule, see here. since there is no detail explanation about destination station, see the station name of the city you are from and to. example : from jakarta (station GAMBIR) to Yogyakarta (station Yogyakarta). OR Jakarta to Surabaya (Gambir to Gubeng).
Bus (inter city) :
There are a lot of bus connected the cities in Indonesia. Mostly the buses are simply find in bus station in the city where you are now OR you can buy tickets in the travel agent in Bus station. Choose Super Executive or Executive class for bus with AC. There are buses with non AC available too. depend on your budget. Several bus i know that i have travelling with :
EKA : Yogyakarta - Surabaya return. AC and comfortable (non AC option is available).
Bintang Prima : Makassar - Toraja return. AC and comfortable. better to book by phone first. see my post here. i think this bus is available for cities in Sulawesi.
Kramat Djati: Yogyakarta - Bandung return. AC and comfortable. i think this bus is available for cities in south of Sumatera (Palembang), Java, and Bali.
Safari Dharma Raya : Bali - Yogya. AC and comfortable. this bus is available for Java and Bali.
i hope this information help you while travelling in Indonesia.