Resting is a very important element of the Torment: Tides of Numenera game. Unfortunately, in the "starting" city of Sagus Cliffs, it could be considered a luxury, which you can only afford after spending a substantial amount of shins (the team cannot rest in random locations within the game world).
Here is a list of steps, which you should take in order to unlock a free resting spot in Sagus Cliffs:
1A) Get to The Changing God Cultists camp in Circus Minor (M3.2). During your first conversation with Casmeen and Mimeon choose the following dialogue options and wait for the moment when you are asked whether you are The Changing God or not (this moment is shown on the picture above). Choose the first option to confirm this fact. From now on you can rest there for free.
1B) During your first meeting with the Cultists tell them that you are not The Changing God (the second dialogue option). If you want to get the free rest option you must make a few steps, described below (2-5).
2) Complete the Fallen to Earth main mission, the last objective of which is to repair a clock in the encampment of the Cultists of the Changing God on the Circus Minor (M3.2).
3) Talk to Casmeen and Mimeon (they are standing next to the clock) and express your willingness to join the Cult of the Changing God. Thanks to that, you will unlock the side mission Disciples of the Changing God.
4) Complete the newly acquired mission - Disciples of the Changing God. It concerns identifying another castoff, who is unknown to the cultists.
5) After the mentioned quest is completed and you will join the brotherhood, ask for an opportunity to rest once again (as shown in the picture above). From this moment forward, you will not be required to pay any fees (you can rest as many times as you like, with any number of characters in the team).