Do I need my own agreement with the airlines to sell tickets through the flight booking API, or does that come through you?

In most cases no, you do not need your own agreement with each airline. The flight content reaches you through supplier connections that are already in place, and you sell under those, so you can begin taking bookings without approaching airlines one by one yourself.

There are real limits to this, and they depend on your market and your plan. Some countries expect a travel business to hold a local licence, or its own accreditation, before it can issue tickets in its own name, and that is a rule you have to meet no matter whose API you use. On top of that, some airlines only release their best fares to agents who hold a direct deal with them, so those fares simply will not appear in your results until such a deal exists. Which suppliers are switched on for you also depends on the plan you are on, so two agencies can see different results for the same route.

What your own airline deal changes is mainly commercial rather than technical. A direct deal can give you better fares and a better margin on the routes you sell most, and it is normally added alongside the connections you already have rather than replacing them. Until you reach that stage, you trade on the connections that come with the service, which is how a great many agencies run for years.

The sensible next step is to ask which suppliers are enabled for your plan and to check what your own country requires before you issue tickets, and if there is one airline or one route you sell heavily, raise it early so a direct deal can be looked at for you.