From my perspective and in my experience, the " 'step by step' directions" that you are seeking for the gift recipient would be more productively directed to the gift giver. Why not nurture the rooted cuttings for a year, growing them into healthy one year old trees that are ready to ripen existing fruit for anyone willing to simply water and sun the tree, and then come winter store it in a cool and preferably dim place, and come spring fertilize it?
In my experience, the best fig tree gifts are given with fig fruit in mid growth that will ripen in the next month or two for the recipient, upon good watering and good sunshine. The ripe fruit will then encourage the recipient to seek out care info for it, or it likely won't matter. Most of the care for the next year or two of a great fig tree gift could already be achieved, built-in, by the gift giver. Doing so gives exponentially greater odds for success, which no "step by step" guide could pretend to match.
It's asking a lot, arguably too much, of anyone with limited growing experience to give them a rooted cutting of a fruit tree and then expect them to be able, however willing, to bring it to fruition in subsequent years. Would seem to be a much more likely prospect of success, if the giver provides the first year or two of work for them in advance, and builds in most of the needed work even for the following year or two.
For that next year or two, for the most part, the gift recipients can push the cruise control button and harvest fruit. Then the giver can at any point convey what was done successfully in the growing, and the recipients can carry on from there if they enjoy the fruit enough to want to fully care for the tree going forward.
Regardless, a few simple suggestions can probably get most people started (these are my current preferences and views; others may differ and reasonably so):
1) Pot Size And Quality: It's a tree, so a rather big pot is a good idea, something at least a foot wide or more, and ideally wider than deeper (to maximize root area exposure to air).
2) Growing Medium: For any new potted fig tree grower, one could either research making many types of useful potting mixes from scratch, or simply and far more reasonably buy one of the best pre-mixed potting materials: Pro-Mix or something like it, or even generic potting mix.
3) Aerated Roots And Moisture With Observant Watering: The Roots power a tree and need access to air in the growing mix as much as they need access to moisture, as much as leaves need access to sunlight. So the growing mix in pots should drain any flow or overload of water while retaining both moisture and air in a spongy springy mix. Observant watering aims for a spongy growing mix, not soggy or constantly swamped and therefore airless. Periodic brief flooding of pots is good to wash out any built up salts but the growing medium should drain well and soon return to a spongy well-aerated state.
(That said, some homemade gravel mixes will not be springy for obvious reasons but they do encapsulate in a hard form the built-in air niches that are characteristic of softer spongy and springy growing mediums that are usually peat-based or bark-based.)
4) Sun And Nutrition/Fertilizer: The tree needs as much sunlight as it can get without wilting badly; it should avoid heavy and steady winds as much as possible to avoid drying and other stress; and it needs some quality nutrition: usually fertilizer in the spring and also in the early and mid growing season in very dilute amounts with each observant watering ideally.
(Also, I smell the growing mix in pots. If it smells sour I add a dressing of lime to sweeten the soil (improving the pH in the way that fig trees like). Happy coincidence that I would rather be around sweet soil, and spongy springy soil, and bright sunlight than the alternatives, and so would figs and their trees.)
How to determine when a fig fruit is ripe enough for the picking is another matter widely written about. Basically if a fig fruit is heavy enough to be drooping on its stem, if it feels heavy and has both swelled and softened, then pick it and see if you like the taste. Experience will soon hint to you whether to pick earlier or later. Ripening time will vary some from cultivar to cultivar, or even year to year, and also within season depending upon the amount of sun and rain and heat. Productivity and time of ripening will also vary by cultivar and by age of the tree and also by the type and quantity of fertilizer including lime provided. Many such variables, it seems to me, go beyond what can be conveyed in gift giving, especially of a rooted cutting. Not that that is a small gift, but it can carry with it many burdens. The more age and experience and durability that can be built in to a gifted fig tree, then the far less information that needs to be conveyed, especially initially, and the far greater chance of that future fig tree and gift recipient's success.
When it comes to fruit tree gifts to novice growers, it's all about the base. The base, the root situation, is the most valuable part. The wood above is also important though secondary. If you can give a tree with a base that is more-or-less set for the next year or two, then you've made theory real. You've eliminated multiple burdens and hurdles. You've taken any step-by-step guide on a piece of paper and created it for real over time in a pot and have essentially projected it into the future tree. It's a lot easier to explain to someone what they are looking at and will be looking at than it is to explain to them theoretically what to do over time in too many ways.
I don't mean to discourage gifting rooted cuttings to novice growers, which might actually be efficient in that many could attempt to learn together, speeding up perhaps the inevitable learning process of trial and error. Quite a lot though would depend on the recipients. I do mean to point out that it could be much easier on the recipients to delay the gift a year. In that case, much more would depend on the already accomplished fact of the year old fruiting tree than on the recipients. That has been my experience at least in gifting trees to friends and family. And also in receiving them, especially when I knew little about viable potting techniques.
And finally for best success in fruiting and growing, I drill holes in the sides of the bottom half of the pots and bury that bottom half of pot in loose soil and mulch. The roots escape the pots to seek further air, water, and nutrition in the ground, and the trees and fruit flourish. I prune the roots with a spade in fall and store the pots in attached garage. This does create a messy pot for storage, which I don't mind. In these pots so far the most productive and earliest ripening types have been Improved Celeste, LSU Purple, and the Mount Etna strains. I expect additional cultivars to match those in the future. Bigger pots that I sit on a cement drive area do not get extra holes, so the size of the tree and crop are more limited. (Self Irrigating Planters would probably be more ideal for this situation.) Of these pots I've found the Mount Etna strains to be most productive so far: Marseilles Black and the like.