Basic programming - Firstly, you will need to have a good grasp of programming. Pick any language you like and go over the basics. This will take you a couple of months.

Software Design - Writing functional software is easy while writing functional AND maintainable software is difficult. Knowing the common design patterns is a must. This will take you another couple of months.

Mobile Development - Since you plan to build an app, you need to know the core of mobile development. This includes application life cycle, how different components behave and so on. This can take you from 6 months up to a year. Don’t forget, you will need to pick up the right programming language (Java for Android development and Objective-C/Swift for iOS).

Software Tools - There are a lot of tools and applications you will need along the way. Version Control (Git) to manage the changes, IDE for development, build tools, testing tools, debugging tools and so on. That is an additional month.

Multithreading - An important area that is often overlooked by beginners. An app like Snapchat requires to do several tasks at once (fetch updates from the server, update UI, and so on). To get a good grasp of multithreading, you will need to spend some time. At least a couple of months.

Networking - Sending and receiving messages is in the core of Snapchat. To be able to build a communication app, you will need to cover the basics of the communication protocols and understand how an app talks to the server. From a couple of months up to a year.

Server-side programming - As a user, all you see is an app, however, there are hundreds of servers behind the scenes that constantly communicate with the client (the app) to ensure your content is up to date. It is important for you to have a knowledge of what it takes to build a responsive server. Another half a year to a year.

Scaling - It is not an easy task to ensure that 150 million daily users can send and receive messages without failures. Each server receives millions of requests every second and it is important to ensure that they are all handled successfully. To do so, one needs to design a good architecture. This is the most sophisticated skill and will take you years to learn.


#NOTE:
Now all that I described above is what you need to learn. What about the actual coding process? I believe it will take you at least 3–6 months (full time) to build an app with the basic functionality of Snapchat assuming you know all of the above areas very well.