How did Gandalf conceal Narya? Where did he keep it hidden? Or did he even keep the ring with him at all?
Short answer: It's unknown
Almost no information is given about Narya when Gandalf comes into possession of it, and even less about how he wore it. There seems to be a reasonable consensus online that the 3 were somehow able to be worn invisibly and only revealed to those the 3 wearers wanted to reveal them to, although I couldn't find any absolute evidence.
The only time Narya is shown to be worn by Gandalf (and coincidentally Vilya by Elrond, Nenya gets another mention too) was at the Grey Havens after the destruction of the One Ring. This leads to speculation that they are now visible to Pippin, Merry and Sam (who's recollections led to the ending of the book) because they have lost their power and can therefore no longer be hidden by their wearers.
Elrond wore a mantle of grey and had a star upon his forehead, and a silver harp was in his hand, and upon his finger was a ring of gold with a great blue stone, Vilya, mightiest of the Three.
Then Círdan led them to the Havens, and there was a white ship lying, and upon the quay beside a great grey horse stood a figure robed all in white awaiting them. As he turned and came towards them Frodo saw that Gandalf now wore openly upon his hand the Third Ring, Narya the Great, and the stone upon it was red as fire.
On her finger was Nenya, the ring wrought of mithril, that bore a single white stone flickering like a frosty star.
The Return of the King, Book VI, Chapter 9: The Grey Havens
There is no reason to believe that Gandalf ever wore Narya on his finger, or that any of the Wise did so regularly. It is only believed to be true as that is what is commonly done with rings and seems to be true (for the most part) for the One Ring. However even while not wearing the Ring, Frodo had a heightened perception and was able to dominate the will of Sméagol to some extent demonstrating that while not wearing it he wielded some of it's power. As such it is possible Gandalf carried it around in his pocket throughout his adventures, or in a hidden section on his hat, or possibly with his pipe!
As for why there is speculation to the wearers being able to hide their Rings, the most common reference is to Frodo seeing Nenya on Galadriel's finger in Lorien (and Sam seeing something but not being able to understand what he saw):
She lifted up her white arms, and spread out her hands towards the East in a gesture of rejection and denial. Eärendil, the Evening Star, most beloved of the Elves, shone clear above [...] Its rays glanced upon a ring about her finger; it glittered like polished gold overlaid with silver light, and a white stone in it twinkled [..] Frodo gazed at the ring with awe; for suddenly it seemed to him that he understood
Yet even so, as Ring-bearer and as one that has [...] seen that which is hidden, your sight is grown keener. You have perceived my thought [...] And did you not see and recognise the ring upon my finger? Did you see my ring?’ she asked turning again to Sam.
‘No, Lady,’ he answered. ‘To tell you the truth, I wondered what you were talking about. I saw a star through your fingers.
Fellowship of the Ring, Book II, Chapter 7: The Mirror of Galadriel