On the face of it, Foraging simply involves walking around the forests and beaches until you spot items you can pick up, and eat, craft or sell. However, Foraging also includes woodcutting - you gain Foraging XP for cutting down trees, and you find cutting down trees costs less energy after you’ve gained more Foraging levels.
Walking around looking for valuables takes time, but no energy; it’s a very useful thing for an early character to be doing.
When the Living on the Land program comes up on the TV, it will give you a tip about the game, which is often relevant to foraging - so, for example, the beginning of each Berry season will be mentioned. There are great numbers of berries to be found in Cindersap Forest (south exit from your farm) during the 4 days of each berry season, and they make good fuel for early trips to the mines. (Berries are usually found elsewhere too, but there aren’t the sheer numbers of bushes along the roads as there are in Cindersap Forest.)
Cutting down trees requires fewer swings when you have upgraded your axe. You will need an upgrade to a copper axe to clear Hardwood Stumps and another upgrade to a steel axe to clear Hardwood Logs. (Naturally these also yield Foraging XP.)
Hardwood becomes quite useful in the later game, but unless you have chosen the Foraging map, you will not normally get more of it from your farm. (Lumberjack skill at Foraging level 10 will also yield some Hardwood from normal trees.) You will need to find your way into the Secret Woods. There is a Hardwood Log in Cindersap Forest which you will need to cut down in order to unblock the entrance to the Secret Woods. In the Secret Woods you will find 6 hardwood stumps you can cut; these renew every day. Four of them are obvious but two are hidden. The Secret Woods is also a good source for foraging mushrooms, and fiddlehead fern.
It is possible to find that the trees on your farm don’t satisfy your timber needs. You can also cut trees not on your farm. You can’t cut trees in Pelican Town, and you can’t cut trees of types that won’t grow on your farm, but Pine, Maple and Oak trees can be cut on the other maps. You can’t plant seeds outside your farm, but new trees will gradually regrow in exactly the same spot - you can cut them again a season later. Now that’s using the forests for what they’re good for!
Rubble gradually spreads in the forest just as it does on your farm. If you’ve got time and energy, it is good to clear up fallen branches and logs, and cut the weeds - this leaves more room for the game to spawn the things you’re interested in collecting. If you’ve only got a little energy to spare, clear isolated branches and logs; cutting weeds with a scythe is still free.
Which skill choices should you make, at level 5 or level 10? At level 5, you have a choice between Forester (wood sells for more money) and Gatherer (sometimes when you forage items, you get a double harvest.) Wood’s really useful, so there’s little point in selling it; Gatherer is obviously attractive when you’re still finding that forage contributes noticeably to your income. The level 10 choice is more powerful. Hardwood is not common unless you’re on the Foraging map (Forest), which makes Lumberjack attractive - for that you need the Forester skill at level 5. (Tapper, the other L10 choice for a Forester, means that syrups sell for more - but who’s selling? Syrups are important crafting ingredients, and you can use all the Oak Resin you can tap.) As a Gatherer your level 10 choices are Tracker (which marks forageables on the map) and Botanist (which means your forageables are iridium quality.) The really interesting thing here is that Truffles, nosed up by your pigs, are forageables. If your pigs are running wild around your forests, Tracker may help you locate all their produce - but Botanist will mean that it’s all iridium quality, which also means it’s so valuable there’s no profit in producing truffle oil; just collect and sell, for easy money! And if you fence the pigs in, the truffles aren’t hard to find. Botanist also means that the forageables you encounter along your way will all be the same quality, so they won’t clutter up your backpack as badly.
At Foraging level 3 you learn the recipe for making Tappers. Initially you want to make three tappers and put one on each type of tree - Pine, Oak and Maple - preferably trees that you often walk past, so that you’ll notice when the syrups are ready. The Community Centre bundles benefit from two of each syrup - some for the Exotic Foraging bundle, some for the Bulletin Board.
Later, you’ll want several maple syrup for a honey farm (plant flowers around a sprinkler & cluster Beehouses around that, for valuable honey about half the year), several Pine Tar for Looms, and all the Oak Resin you can possibly get your mitts on, for Kegs. This means that you want to plant more tappers on Oak Trees once you can readily spare the ingredients, even if you’re saving up big piles of Oak Resin. In year 3 it’s not unreasonable to want 100 kegs.
You’ll often have part of your farm cluttered with trees, until later in the game. Visit this area occasionally (or scan it when you’re looking for a place to put a building, at the Carpenter’s shop - you can look over your entire farm this way!) and keep an eye out for a Mushroom tree, which are sometimes created during Fall. If you’re lucky enough to get one, put a tapper on it, and clear the space around it so that it can spawn saplings … it will produce mushrooms every week or so, and just sometimes these are red or purple mushrooms.