Just started my first AD&D 2nd edition game and I'm a Druid. I want to gain an animal companion but I don't know what's involved. Do I need certain proficiencies like Animal Handle and Animal Training? Am I only limited to the spell Animal Friendship? And how am I going to make friends with the woodland animals like wolves, bears, and spiders? Won't they just attack me next time they see me? I know Animal Friendship is permanent unless left alone for a week but isn't there a way to befriend the animals near my Sacred Grove without the HD limitations of the spell?
Gaining an animal companion wasn't a thing druids could do any better than any other class in 2nd Edition, at least not without using kits* and other options from non-core books.† (This was more the province of 2e rangers, who acquired animal companions instead of human followers at higher levels.)
To get an animal companion in 2e as a druid you had to do it the hard way, by roleplaying out the relationship with the animal and taking the risks inherent in dealing with wild beasts (which are somewhat easier to mitigate as a druid). This is much like how you get ahead in every other aspect of the game though: you play it out and reap the rewards of your efforts, rather than have things just kind of happen as you level up. Pursuing goals in that way is definitely an adjustment coming from later editions, but it's satisfying.
Gaining an animal companion in the 2e style of game rules, then, is just like making friends with any other NPC, except for the communication challenge. Do things that will make them like and trust you, and don't do things that will make them dislike you. Animal friendship is a useful spell for getting past the initial hurdle of making that first connection. Animal Lore is useful to understand animals' behaviour so you can act appropriately. Insist that your DM uses Reaction rolls, so that not every beast you meet is automagically hostile to you; when you meet a friendly animal, build that connection to try to make it lasting.
Once you have befriended an animal enough that it now considers you a friend and companion, Animal Training can help to teach your new friend tricks and skills, but with the caveat that the skill as written only works on one type of animal, chosen when you take the skill, so it's unlikely to be useful for whatever kind if animal you've befriended. Animal Handling won't be useful unless you have befriended some sort of pack animal; I suppose meeting a friendly feral donkey is a possibility, but I wouldn't choose skills based on that chance.
Finally, realise that an animal companion is a friend who has chosen to accompany you as an equal. They will only fight if it makes sense for them to fight, so they're not the disposable muscle that some players and DMs treat them as in later editions. Because of this, don't overlook the helpfulness of having small friends in high places – an owl that spots an ambush is as useful, if not more so, than a wolf who might only die if they tried to protect their friend from a couple of bandits.
* The Beastfriend druid kit from The Complete Druid's Handbook gives a druid bonuses to Reaction rolls with animals, but still doesn't have a "and now you have an animal companion" mechanic. It simply states that Beastfriends tend to befriend animals and accumulate friends and pets, and even that is put firmly in the domain of roleplaying it out.
† The "2.5e" book Player’s Option: Skills & Powers offered elves an optional racial feature that gives a bonded animal companion. It also included a kit, the Animal Master, that included a bonded animal companion.