Update: Rehauled for concision and simplicity.

If you have interest in gardening, feel free to spill out here. With personal housing coming up, and finally (probably) enough research, this thread will serve to give a general guide and outline of gardening in the game.

Gardening is simple: To plant a crop, you need 1 soil and 1 seed. You can gather seeds as a botanist, and you can gather soil as a miner. During the growth process, you can fertilize the crop, using an item called Fishmeal.

Currently the FC has 3 deluxe patches and therefore have 24 plots total. Small houses can only have 1 patch, medium houses 2 patches, and large houses 3 patches.

What seed to plant? What soil to use?

About Soil

The descriptions for the soil items tell you what they do.
Shroud Soil: For quantity.
La Noscean Soil: For quality.
Thanalan Soil: For interbreeding.

However, soil comes in 3 grades. Grade 1 is harvestable at level 26 from level 30 nodes. Grade 2 is harvestable at level 46 from level 50 nodes. Grade 1 and Grade 2 appear randomly in different slots at the crystal/carbonized matter nodes, with varying rates of appearance. As of the edit to this post (8-22-2014) the Grade 1|2 soils are selling decently, so gather if you see them.

For the record, before I was geared enough to get Grade 3 soil, I spiritbonded non-melded miner gear entirely just trying to get Grade 2 soil, and needless to say I got maybe like 16 soils and hundreds of carbonized matter, enough that I made a lot of money off the matter and enough that I got the 4000 mined achievement in Thanalan. I recommend gearing up for Grade 3 if you are a 50 Miner because 1) you know for sure you're getting a soil even if it's about 1 per hour and 2) it has the highest 'probability' for soil of every type.

tl;dr Grade 2 soil is the Atma of Mining. Gear up for Grade 3 if you're serious about gardening.

Grade 3 soil requires 370 Gathering stat on your Miner before you can obtain it. The node spawns in the vicinity of the Silver Bazaar in Western Thanalan at 5AM Eorzea Time, and the soil is always in slot 8 (Water Cluster in slot 5 if you feel like picking that up too).

Why Grade 3 over Grade 2?

For interbreeding. When you use Thanalan soil and plant two different seeds next to each other, they have a chance of crossbreeding and producing a seed of a completely different crop. The higher the grade of the soil, the higher the chance of interbreeding.

If you know the combination for your desired crossbreed seed, it's possible to getit by using Grade 2 Thanalan soil. But this hasn't been thoroughly tested.

The amount of time it takes for the crops to grow range from 3 to 11 REAL WORLD DAYS.

Currently the useful soils to get are Thanalan and Gridania. The crops that grow in 3 days are chocobo foods and snacks. The crops that grow in 6 to 11 days are the deluxe items (Glazenut etc.) and Thavnairian Onion.

Largely, the soil that produces seeds is Thanalan soil. There is a very, very slim chance that a crop planted on a non-Thanalan soil will produce a seed, but it's chiefly the Mandragora minion seeds and it is an extremely slim chance.

With the introduction of chocobo food/snack seeds, RNG has been introduced in the combination results. Most older combinations are not affected by RNG; chocobo food seeds however have a chance of producing a different crossbreed seed even if both are planted with Grade 3 Thanalan soil.

tl;dr Chocobo food sucks

About Seeds

The amount of seeds available grows with each patch update.

Research reveals that if you want the deluxe seeds (Glazenut, Broombush, Jute, etc.), then it follows that you should be using the higher level seeds. Think about the seeds in terms of what level you can gather them as a botanist. Rolanberry, Almond, Linseed, and Mandrake, which are relatively higher-level gathers, are used to produce some of the 'lower' heavy-duty crossbreeds like Azeyma Rose, Apricot, and Shroud Tea. Those in turn are used to obtain deluxe seeds.

Retainer ventures are one means of obtaining the currently unobtainable seeds, like Broombush and Glazenut. However, they can only be planted once and harvested once, and after that we have no means of producing more unless the method of crossbreeding such seeds is found. Moreover, even if planted on Gridanian (quantity) soil, it is likely these seeds will still yield only one item, compared to the yields of lower level plants (5, 12, etc).

The one-star seeds are obtainable at level 50 unspoiled nodes with only about 320 gathering. No need to meld to get those.

Best Practices

The resulting crossbreed seeds are determined from the moment of planting. If you plant a seed when both plots next to it are empty, you will not get a crossbred seed.

Ideally you plant when there are crops on both sides of the empty plot. If you are starting with an empty patch, plant clockwise with alternating seeds.

A method recommended by the gardening community is to have an additional spare soil and an additional seed of the two seeds you will alternate to crossbreed.

Example: For Nymeia Lily seeds, obtain 8xGrade 3 Thanalan soil, 1xRandom soil of your choice, 5xAlmond seeds and 4xMandrake seeds. In the empty patch, pick a corner plot and plant 1 Almond seed using the 1 Random soil. Next to it, plant 1 Mandrake seed with Thanalan soil, then alternate Almond and Mandrake in that direction until you reach the first plot again. Remove the "1 Almond on 1 Random" crop; replace with Almond seed using Thanalan soil.

After the initial empty plot planting, it is easier to crossbreed if 'corner crops' and 'center crops' are harvested at different times. Given different times of maturation for different crops, this is fairly easy.

Example: Glazenut seeds are obtainable from crossing Apricot with Glazenut. Apricot matures 1 day earlier than Glazenut does. Apricot can be harvested first and then replaced with Apricot in the same spots, since the Glazenut plants are already present and adjacent to the empty plots.

Moving on to the growing process

After you plant a crop, when you select the crop again it will open a menu that includes the options "Fertilize Crop" and "Tend Crop".

Tend Crop

This is a basic action that keeps your plants in a "doing well" state, and only needs to be applied about once per Earth day. Even non-FC members can tend crops, so I assume come personal housing if you are busy for a few days, you can ask a buddy to tend them for you. If crops are not tended for over a day, they get a purple and yellow smoke around them indicating unhealthiness. The crop can stay unhealthy for a few days, but it's likely they die after an extended period of time. 'Unhealthiness' can be cured just by tending the crop once.

There's two different animations for crop tending depending on rainy weather and not rainy weather, so don't worry if it seems like you haven't tended the crop just because you weren't using a watering can.

Fertilize Crop

The fertilizer is an item called Fishmeal. The Material Supplier in the housing areas currently sell them for 21 gil. Level 1 Culinarian can also make 12 Fishmeal a pop using 3 Lominsan Anchovy, which is fishable by level 1 Fisher and caught using the Lugworm bait (1 gil each from Fieldcraft Supplier in Limsa). If you don't mind netting other stuff while fishing, then Limsa and Middle La Noscea are viable places to catch Lominsan Anchovy. If you want to fish Lominsan Anchovy only, you fish at Candlekeep Quay in Lower La Noscea.

A crop can be fertilized once every Earth hour. The actual effect of fertilizer is unknown; the only discernable result is faster growth, so it's relatively advantageous to use fertilizer on your crop as much as you can.

About Patches

The Junkmonger downstairs and the Housing Merchant in the housing area markets sell garden patches, which come in 3 sizes. The one shown here is a deluxe:

If you know which crop in particular you can gather seeds from (in a Prickly-Rolanberry combo, only the Rolanberry plant produces crossbred seeds), then you can come up with particular configurations to produce seeds. (See here for an idea of patch config.)

Resources

FFX|V Gardening is now THE go-to site regarding gardening crossbreeds.

Official forums thread
Might be useful. I used it to spy on what other servers were doing.

Reddit super spreadsheet
Despite the usefulness of this spreadsheet, it's editable by anyone and therefore some entries are possibly fake. However with good guessing it's possible to come up with viable combinations.

Scroll down past that silly British guy to see what gardening products are used for and for general combinations if you are too arsed to click on the link above to get the FFX|V Gardening.

Additional Questions

What effect does Grade 3 soil have on the deluxe seeds, like Glazenut, Jute, etc.? (courtesy of Scything Moon)

Currently (8-26-2014) Grade 3 Shroud soil is useless for Glazenut, Jute, Broombush, Thavnairian Onion, Umbrella Fig, and any other seed in the same tier. Regardless of Shroud soil, those seeds will yield only 1.

Grade 3 La Noscean soil may find use for slightly lower tier seeds, like Dalamud Popoto, Apricot, Royal Kukuru, Star Anise, etc. due to the fact that these products are used in master culinarian cooking. However, the two-star foods have yet to see much use in endgame. Therefore, Grade 3 La Noscean soil is not very useful. Moreover, the Glazenut/Jute/etc. seeds are used in crafting recipes that do not need HQ, rendering the use of this soil worthless for that set of gardening products.

Grade 3 Thanalan soil has the most use for the highest tier of seeds. The high tier seeds are still able to cross with other seeds to possible produce other high tier seeds or even self-perpetuate.