If so, does that mean that it can learn skills? Would I get to choose how I want it to level up?
Animal companions don't gain XP or level up in the same way that players do.
That said, they do gain hit dice (HD) as your Druid level goes up. Those are listed on the table for the Druid's Animal Companion. Each HD is roughly equivalent to a level. They gain HP from their HD and Constitution, skill points, ability points every 4 HD, and feats every 3 HD. The amounts of skill points and which saves/BAB to use are listed here, but animal companions never get the size increase listed for animals, nor are they HD capped. This answer has some more details.
In addition to that, they also gain the other bonuses listed on the Druid table, depending on your Druid level. This can make for a choice between getting a newer and more powerful companion, or levelling up the one you already have. A stronger companion lowers your effective Druid level, so it would get fewer (or none) bonus HD and other bonuses on the table. Which animals are options and what your effective Druid level becomes for them is listed on the animal companion link above.
I've found that a simple way to think about it is that when your Druid reaches the appropriate level, your companion gains two levels in the "animal" class.
Note that as they use natural weapons, animals don't gain more attacks from higher BAB. The Druid "Multiattack" companion bonus can give a companion with one attack a second one, or improve it's attacks if it has multiples. Otherwise the simplest way to get more attacks is to switch to a more powerful companion as your level goes up.