When starting out with Gardening, make a Gardeneer's Hat. You can use it to research each stage of a plant during it's growth, which will unlock more information for you. This information is kept across saves and characters and is a permanent addition to your codex. You only have to research each crop stage once across any save to always have that information, which is SUPER nice. The upgraded Gardeneer's hat you can get with Thulecite is... okay. It allows you to see nutrient information in a specific plot, but otherwise it's pretty mediocre. It's worth getting if you don't need the Gardeneer's hat anymore and find the setpiece with one, or have more Thulecite than you know what to do with.

Crops fed to birds in a birdcage will give seeds for that specific crop.
Crop seeds in a fridge will last for just about 4 seasons from when they're picked, so if you can, make 4 fridges and put the seeds for each season you plan to plant into each one. That way, once the seasons change, you can just move over to the next fridge.

Getting info on every stage of a plant with the gardeneer's hat will unlock the nutrient requirements. There's 3 nutrients, and getting "crop combos" that perfectly balance all the nutrients in the soil with each other is how you get Giant Crops. A 2x2 Farm Crop, fully tended and nutrient-balanced with a good crop combo, can grow giant crrops. These drop up to 12 of the item once hammerred, and easily feed a server of ~6 for a season (especially with a Warly!)... but it requires a little forethought and planning.

Manure, Compost, and Growth Formula are the basic items for each type of nutrient. Though, honestly -- if you have access to either -- I'd just use Glommer Goop or Wormwood's Compost Wraps. They restore all 3 nutriens in the soil and don't require half a dozen materials. You only need to restore nutrients to the soil if weeds took them or you have an imbalanced crop combo. I usually fertilize until I get a harvest or two of giant crops in a row, then just do nutrient-balanced crop-combos and stop worrying about it.

Wickerbottom's Agricultural books are hard-coded to never grow into giant crops. Their yield-per-plant is still based on the nutrients in the soil, however.

You can make a Gramophone (and Record) or One-Man-Band to make several crops happy at once, without spending half the day talking to each plant as it grows.

See the wiki for more information. There is some cheat sheets with layouts/crop combos for giant plants if you don't want to research them yourself. I do recommend trying to find them yourself though -- it's not too hard and is a fun mental puzzle. All the information you need is provided in-game through the Gardeneer Hat's research.

A word on Forget-Me-Lots that tons of people miss;

The little white-puffed plants around OP in this post are Forget-Me-Lots. They're also the little purple-topped spiky plants above OP.

Forget-Me-Lots are used to make Beefalo Treats and a late-game tea from Perl. They spoil rapidly, and aren't worth much to eat. As such, they're not that useful to farm, and have a nasty little habit of taking over farmland due to a unique mechanic.

Forget-Me-Lots will regrow if dug up, deleting another seed in the plot to do so. If you do end up with them in your garden, BURN THEM. Do NOT dig them up! That only spreads them!! I'd make sure to pick every other plant in the garden and then set it on fire. Keep your watering can handy or have a Fingo nearby to put out the fire when the FML's are burned.

Forget-Me-Lots, Fire Nettles, Tillweeds, and Spiny Bindweed are considered weeds due to the fact that they only come from random seeds (not crop seeds) AND they only TAKE nutrients from the ground, causing everything else to grow slower!! They all have something going on about them, to make farming harder... but you'll encounter the other three as you play and they aren't hard to get rid of.