You can see a list of all changes in the "What's New in TypeScript" document on GitHub:
https://github.com/Microsoft/TypeScript/wiki/What's-new-in-T...
And a list of breaking changes here:
https://github.com/microsoft/TypeScript/wiki/Breaking-Change...
They both go back until v1.1. For older changes you can check the blog:
https://devblogs.microsoft.com/typescript/announcing-typescr...
https://devblogs.microsoft.com/typescript/announcing-typescr...
Anyway, it's hard to say what you need to do to get your code to compile on the latest TypeScript version without knowing what some common compile errors are in your code.
Here are some guesses:
1. The way you obtain type definitions for third-party packages has changed. It used to work with /// <reference> and tools like nuget or TSD (TypeScript definition manager). But now it works with npm in the @types namespace and some npm packages even include definitions themselves now.
Of course, updating to the latest type definitions would mean that you have to upgrade to the latest versions of the third-party libraries as well. Otherwise you have to find a way to keep working with the old definitions for the respective library versions.
2. Pre-1.5 TypeScript had something called "internal modules" which were renamed to "namespaces" and are generally discouraged now. Hopefully your code is not using that feature.
https://www.typescriptlang.org/docs/handbook/namespaces.html
3. tsconfig.json. I don't remember when they introduced it, but you likely need to create this file and tweak the settings