How do I set a fleet to orbit a planet?

No you're doing it right.

You just have to right click on a planet you own to have it orbit. If you have modules like Crew Quarters or Engineering Bays then it will reduce the cost further.

I'd check again to make sure because it really is just as simple as right clicking a planet you own - if you don't own it you won't get the reduced cost.

The reason transport ships have the option when you right click is because they can land on planets unlike your military fleet.

Fuller: Right clicking on the system will just move it straight to the system. If you go into the system, go to the planet that you own, and while your fleet is selected, right click on the green bar with the planet's name, your fleet should travel there and enter orbit.
You can confirm this by looking at the fleet and near the Admiral it should say: "Orbiting 'Planet Name'", also your income should go up as maintenance is lower for being in a planet's orbit (with a space station).

Right click the space station, not the planet.

Technically it is the space station with its modules what reduces the maintenance cost. This also explains why the ships entered the low-maintenance mode after upgrading or healing.