How to implement GPS tracking cheaply
at this point just buy one of the dog trackers for $60, I dont think you are going to make one cheaper.  Unless you buy a cheap/used smart phone and activate it and leave it with the bike.

If you want to do it yourself you will need a gps receiver, pretty cheap like $10 on ebay.  Super easy to parse the NMEA data.  You will need some sort of processor, a microcontroller will do, or a raspberry pi zero or something like that.  You will need power.  And if you want to track it with your phone you will need a cellular module, that is going to cost several times the cost of a used phone.  YOu might find some on ebay, but note they are not necessarily what you think they are.  Adafruit has one, you will need a sim card and service for it.  And then work out how to talk to it, for texting or other such things it is not too bad, there are libraries.

You are better off activating a cheap or used phone and running an app and keeping it charged.