Here are the mods I recommend. These are all compatible with the latest versions of SMAPI and Stardew Valley on Linux/macOS/Windows. See the player's guide to using mods if you're interested.
Quality of life
I think the best mods are those which improve the game while maintaining its balance, lore, and style.
AutoGate Gates open automatically when you approach and close behind you.
Better Sprinklers Customise the sprinkler radius, with a proportional change to their cost. If you're willing to mine all the ores you'll need, that makes the basic sprinklers useful early in the game and lets you do more than just watering crops every day.
Billboard Anywhere View the calendar and daily quest anywhere. See upcoming birthdays and festivals without needing to walk to the nearest calendar.
Chests Anywhere Access, label, and categorise your chests from anywhere (with optional balanced mode). Makes inventory management less of a chore — you can have multiple organised chests and sort things away without walking around trying to find which chest is which.
Data Layers Toggle an overlay to see live bee house / Junimo hut / scarecrow / sprinkler coverage, hidden paths and warp points, and more. Very useful when laying out your crops.
Fast Animations Makes the routine player animations (breaking geodes, eating, drinking, etc) almost instant.
Generic Mod Config Menu Edit mod options in-game without needing to manually edit
config.jsonfiles, for the many mods which support it.
Horse Flute Anywhere You'll often find you left your horse across town, or the horse can't pass through a narrow path. This mod eliminates horse location management as a game mechanic; just play the flute from anywhere and he'll teleport to you.
Lookup Anything Point at almost anything, and press a button to see information about it. This helps you decide whether an item is worth keeping (e.g. it shows what bundles/recipes/achievements you still need it for), plan your crops (e.g. it tells you when a crop will be ready), manage trees (e.g. it tells you whether something is blocking their growth), see what gifts an NPC likes, etc. Has an option to only show gift tastes you've discovered in-game.
NPC Map Locations NPCs move around a lot, and it can be frustrating to find them when you need to talk to them. This mod shows their locations on your map.
Pony Weight Loss Program Lets the horse pass anywhere you can. This eliminates a common annoyance where the horse can't pass through narrow paths.
Quick Glance Press a button to zoom out and view the surrounding area, and release it to return to normal. This lets you quickly get your bearing, look for nearby forage, etc.
Skip Intro Skips the starting splash screens and (optionally) the title screen. It's only a few seconds each time, but you won't be able to go back once you're used to this mod.
Tillable Ground Lets you plant crops (and grass) on the grassy portions of the farm map.
Toddlers Like Parents Your children will look a bit like your spouse (instead of the default generic children). You can optionally customize the clothing they wear too.
These mods are best used later in the game, once you've already played for a while with the vanilla mechanics.
Automate When you're tired of manually working each machine, this opens a whole new gameplay mechanic. Place a chest next to a machine (like a furnace or crystalarium), and the machine will pull raw items from the chest and push processed items into it. Connect multiple machines with a chest to create factories.
Deep Woods Adds an explorable, procedurally-generated forest area to the game with many secrets and resources.
Tractor Mod Buy a tractor from Robin to make managing large farms much easier. The tractor lets you quickly clear debris, plant/fertilise/water/harvest crops, etc.
These mods visually tweak the game using Content Patcher.
Fixed Mouse House Give the hat mouse a less broken-down house.
Natural Clock Change the Gold Clock to fit in better the game style and look a bit less ostentatious.
Rustic Traveling Cart Change the traveling cart to better fit the game style and look a bit less silly.