I am writing a short story about an overworked planet whose civilization has achieved space elevators and orbital rings.

Note that there is no known material strong enough to build a space elevator. People will talk about carbon nanotubes, but even with the best manufacturing capabilities imaginable small imperfections in the material will reduce is breaking lenght to a point to short for elevators. With orbital rings you picked the best lauch assist megastructure there is, since the ring is unchangeable regarding scalability, general utility, throughput and possible take of velocity.

The planet, which is earth-sized, has three complete orbital rings in stable orbit, each one 200 meters in width, which is about an eighth of a mile.

Sounds perfectly reasonable dimension wise. Note that orbital rings have tremendous energy requirements to be started up, a skeleton ring will need half our anual energy priduction to get going as noted is this paper, which will give you more information on orbital rings, but the civilisations building them should have mastered fusion or of world solar or at least mass fission, so it will be fine.

The sole purpose of the rings is to allow spaceships to enter orbit.

As Algebraist already noted is his answer, rings can be used for so much more. Definitly check out the Isaac Arthur video he linked, the guy is amazing. Concerning other things to do with the ring: hyper fast on planet transport, shellworlds, supermundane planets or megaearths, chandelier (hanging) cities (or gardens), colonising ice giants, gas giants and even suns and much more.

Where do you have to put the rings relative to each other for them not to have any issues or break, and should the rings be fixed onto the ground or allowed to orbit freely?

The sinple answers are whereever and whatever you like. As long as the rotor sits in an orbital path around the planet and two rings rotors do not intersect everything is possible. Maybe having one equatorial ring for transfer to the planets habitat swarm and moon, one aligned with the systems plain of the ecliptic for interplanetary launches and one polar ring for power beaming receivers (Atlas towers will do the same) on the poles and more exotic launch trajectories is an optimal three ring setup.

I would suggest ancoring them to the ground, because one would loose so much of the rings utility if they where free floating. The only times you wanna use free floting rings are when you are colonizing objects with no solid surfaces like ice-giants, gas-giants and suns.

I am writing a short story about an overworked planet whose civilization has achieved space elevators and orbital rings.

Note that there is no known material strong enough to build a space elevator. People will talk about carbon nanotubes, but even with the best manufacturing capabilities imaginable small imperfections in the material will reduce its breaking length to a point too short for elevators. With orbital rings you picked the best launch assist megastructure there is, since the ring is unchangeable regarding scalability, general utility, throughput and possible take of velocity.

The planet, which is earth-sized, has three complete orbital rings in stable orbit, each one 200 meters in width, which is about an eighth of a mile.

Sounds perfectly reasonable dimension wise. Note that orbital rings have tremendous energy requirements to be started up, a skeleton ring will need half our annual energy production to get going as noted is this paper, which will give you more information on orbital rings, but the civilisations building them should have mastered fusion or off world solar or at least mass fission, so it will be fine.

The sole purpose of the rings is to allow spaceships to enter orbit.

As Algebraist already noted is his answer, rings can be used for so much more. Definitely check out the Isaac Arthur video he linked, the guy is amazing. Concerning other things to do with the ring: hyper fast on planet transport, shellworlds, supermundane planets or megaearths, chandelier (hanging) cities (or gardens), colonising ice giants, gas giants and even suns and much more.

Where do you have to put the rings relative to each other for them not to have any issues or break, and should the rings be fixed onto the ground or allowed to orbit freely?

The simple answers are wherever and whatever you like. As long as the rotor sits in an orbital path around the planet and two rings rotors do not intersect everything is possible. Maybe having one equatorial ring for transfer to the planets habitat swarm and moon, one aligned with the systems plain of the ecliptic for interplanetary launches and one polar ring for power beaming receivers (Atlas towers will do the same) on the poles and more exotic launch trajectories is an optimal three ring setup.

I would suggest anchoring them to the ground, because one would lose so much of the rings utility if they were free floating. The only times you want to use free floating rings are when you are colonizing objects with no solid surfaces like ice-giants, gas-giants and suns.