Releasing a constant stream of fire takes very little shape manipulation, as it's just building your fire chakra up to the point of release, and then doing so in a flowing fashion. The bet analogy I can think of is like you're filling up a bucket with water, and then it overflows. The point of release would be when it becomes full, and then the release itself is the overflowing of the water. Because this method is typically uncontrolled, you're breathing it out which would be stronger at first, and as you continue it would gradually become weaker.
Running Fire I believe is created from the ground, but I digress.
Creating a stream of fire and manipulating it to form a circle around a target requires a little more work than just releasing it. The methodology is the same, as in you're breathing out flames in a stream, but once it reaches a certain point you'd need to essentially bend the flames once they reach a certain point in order to form a circle. If you're using it on a target, if you aim it directly at them you'd need to manipulate it to circle around them at a much sooner time, whereas aiming it to their side, you can pretty much make it flow around them, which wouldn't require as much effort since it would feel fluent.
I'm not entirely sure how one would go about making streams of fire follow a wire like they were guided. They could be doused in a flammable liquid beforehand which could help it, of course but going about that would be hard. I suppose doing something similar to what Neji did with his Jyuken to attack Kidomaru through his web: firstly infusing the wire with chakra, and using the stream of fire to essentially follow that path of chakra.
Running Fire I believe is created from the ground, but I digress.
Creating a stream of fire and manipulating it to form a circle around a target requires a little more work than just releasing it. The methodology is the same, as in you're breathing out flames in a stream, but once it reaches a certain point you'd need to essentially bend the flames once they reach a certain point in order to form a circle. If you're using it on a target, if you aim it directly at them you'd need to manipulate it to circle around them at a much sooner time, whereas aiming it to their side, you can pretty much make it flow around them, which wouldn't require as much effort since it would feel fluent.
I'm not entirely sure how one would go about making streams of fire follow a wire like they were guided. They could be doused in a flammable liquid beforehand which could help it, of course but going about that would be hard. I suppose doing something similar to what Neji did with his Jyuken to attack Kidomaru through his web: firstly infusing the wire with chakra, and using the stream of fire to essentially follow that path of chakra.