I need to create several GIS based application and embed them on a web page for professional development. I need a cost effective solution and don't have big budget. Any help in right direction and to get started is appreciated (case study etc.).
An answer a bit generic without more details.
An introduction to GTFS to read
You also need to import GTFS in database
For viewing, really depend but if no routing needed, just use you db to make SQL join from tables and export features to geojson (e.g PostGIS) if you need vector.