Google Maps is what I recommend. Sounds basic, but it helps.
I stayed in Myeongdong for 5 days and it helped a lot in navigating the city. It provided enough information on subway network. Thus, going from one destination to another became faster without further browsing.
A friend recommended another app, however I uninstalled it and continued using Google Maps because I already familiarized myself with the information being provided like subway line number, bus number, stops, destinations, etc.
With just the GPS location [GPS on the device must be enabled] and manually entering the destination, the app can provide bus, subway, train, and any other transportation options. The only thing that I hate is it does not provide walk options. CMIIW
Despite that, the GPS pin is helpful in locating my current position in an area. Thus, walking and backtracking are easy. As I move, the pin updates. Cool!
When I went to Korea, I bought a prepaid SIM card with 1 GB data allowance which I used mainly to operate the Maps app. Free public WiFi helped too.