You start Astroneer with a starting set of items. Among them, you will find an Oxygenator, a device that delivers oxygen to tethers. This, in turn, allows you to bring oxygen to any place on the planet. In the beginning, tethers are the easiest way of having constant delivery of oxygen. Learn how to connect tethers to an Oxygenator - this will allow you to explore the world freely.
How to connect tethers to an oxygenator?
The starting set has an Oxygenator. Connect tethers to an Oxygenator to deliver oxygen. However, placing an Oxygenator on the ground won't do anything - the device won't work.
Place an Oxygenator on a Starting Platform to make it work - the latter is also found in the starting set. Starting Platform is for holding not only Oxygenators but also for other objects, e.g. printers creating new items.
Once you place the Starting Platform and install the Oxygenator, you will be able to connect tethers to it. Oxygen will flow through the tethers. This will be indicated by the lines glowing blue. Remember to place an Oxygenator (installed on a Starting Platform) near your base. Otherwise, the Oxygenator will not produce oxygen.
The starting Oxygenator is only one of the few ways of producing oxygen. More complex devices have to be crafted. More information about that in the "Crafting" chapter.