I don't ever really go out to specifically hunt (unless I'm going after some animal I spotted near my settlement), instead I simply hunt animals that I encounter when doing my regular activities like trading exploring, gathering and small game trapping. I then load the meat and furs (or sometimes whole carcasses) on my dogs for further processing.
To find animals, climb on hills and mountains on the zoomed out (world) map. If you don't have any, the lichen forests are the next highest elevation. If you don't have any of those or have very few (Reemi territory and the area around it has problems with too low elevation, notably), I recommend hunting elsewhere. You don't often actually see the animals around from your high vantage point, but you sometimes do and this is the most hassle free way of finding them. If you don't at first see any try the next hillock or recheck ones you have already been to. If you are really desperate to find animals right now, don't just randomly walk around on the world map, but zoom in occasionally to check if you see anything. Almost equally important is picking the right terrain where to hunt: I consider hunting animals in spruce mires and conifer forests act of desperation (excepting bears since they hold aggro well). Open mires on the other hand are great places to hunt as you often find yourself cornering animals against the numerous ponds that dot this terrain and they offer clear lines of fire.
Character wise, speed and endurance are the most important stats, even more so than the weapon skills as active hunting in URW does ultimately come down to some variety of persistence hunting. Some animals like lynxes and badgers can be simply run down with fast, high endurance character and some educated guesswork, even in the summer when they don't leave tracks on most terrain. Also keep mindful of your encumbrance, for example the difference in fatigue buildup between 1-2% and 5-6% is already huge and apparently non linear.
For archery the most important thing to ensure is to have a clear line of fire: don't shoot the pines. If you are after some forest reindeer for example it's a good idea to hang back a bit so that they don't start running and wait for a good opportunity. Though if you encounter the reindeer via the sudden encounter type where the game pulls you in a middle of the herd, don't waste your opportunity for the close range shot. I think shooting (with bows, javelins are short range only) at the max range is not to be underestimated. You'll often miss sure but this often doesn't send the animal running yet and you can make several attempts. Just keep the line of fire clear. Since the changes to cord economy some time ago arrows are really cheap up to whatever level you can craft them at with even modestly successful bird trapping.
There are also some kinks to keep in mind that you can exploit: for example animals often like to run in diagonal zig-zag pattern that you can learn to predict to a degree, and herd animals in particular like to rebound back towards you after they stop running. This is particularly useful when persistence hunting forest reindeer. First you scare the animal, then you alternate between running and walking after it using both your observation of the animal, it's tracks and some guesswork to minimize time loss. You'll soon see the animal again rebounding towards you after it stopped running. Run towards it to scare it again and then resume your usual mix of running/walking and following it. Rinse and repeat and you'll soon have an otherwise unharmed breathless forest reindeer.
Ultimately successful hunting can take patience and the "I think I'm going to give up on this animal. Too much of a bother. But no, not just yet! And now I finally killed it!" is the greatest gameplay experience in URW.
Lastly, if you are getting fatigued from just walking around you'll need a pair of skis and a ski stick, or you'll need to practice your skiing skill. I don't actually recommend starting in spring as new player since it really starts you in the last days of winter, start in summer and autumn instead.