When an animal companion first starts out, it only knows one trick. Does this mean that if I don't choose something like heel for the first bonus trick than I must attempt to "Push" my companion? Since this is likely to fail at a low level, won't this cause my companion to be abandoned or slow down my party because they are waiting on me to try to get my companion to stay with us? I don't understand how I can even begin to control my companion when I am so limited. Am I missing something?
It’s rare for a druid to ever have to push their animal companion. They receive many bonuses to Handle Animal, and it’s usually assumed that the druid has had a lot of time with the animal companion, plenty in which to teach it many tricks.
Total Tricks: not just 1
An animal companion starts out with one bonus trick, over an above however many that animal could be taught if it weren’t an animal companion. See Bonus Tricks:
The value given in this column is the total number of “bonus” tricks that the animal knows in addition to any that the druid might choose to teach it (see the Handle Animal skill). These bonus tricks don’t require any training time or Handle Animal checks, and they don’t count against the normal limit of tricks known by the animal. The druid selects these bonus tricks, and once selected, they can’t be changed.
and Handle Animal:
An animal with an Intelligence score of 1 can learn a maximum of three tricks, while an animal with an Intelligence score of 2 can learn a maximum of six tricks.
So your animal companion actually knows four or seven tricks, depending on its intelligence.
Link: Handle Animal bonuses for your animal companion
Furthermore, druids receive considerable bonuses for handling their animal companion:
A druid can handle her animal companion as a free action, or push it as a move action, even if she doesn’t have any ranks in the Handle Animal skill. The druid gains a +4 circumstance bonus on all wild empathy checks and Handle Animal checks made regarding an animal companion.
Note that if you are riding your mount, you probably aren’t using your move action for anything: your mount is doing the moving, after all. Thus, using your move action to push your animal companion won’t slow you down unless you fail. DC 25 is pretty high at 1st level, even with the +4, but in a few levels it will be pretty easy. And you should not have to push your animal companion often at all.
Training: you usually take-10
Between that, the four ranks you should have, you easily have a +8 bonus to Handle Animal at first level. Maybe +7 or even +6 if you dumped Charisma, but then you might also have +9 or +10 if you invested in it. Anyway, +5 is sufficient to take-10 for training many tricks, and +10 will get you any of them. Since you do not need to use Handle Animal for bonus tricks, you can use the bonus trick to get one of the DC 20 tricks (e.g. Attack or Defend), and get DC 15 tricks for the ones you actually train the animal in.
By level 3 or so you should be up to +10, and thus able to take-10 on any trick training you want to do.
Backstory: this is usually where training happens
Note that most druids start the game with an animal companion, rather than acquire one during play. Usually they have been with the animal companion for quite some time when the game starts, and therefore have had time to train the animal companion in all of their tricks. They may even have had time to train without taking 10, since you can keep trying Handle Animal checks until you succeed (each failure takes half as long as successful training for the trick).
Thus, training should not, in most cases, slow down the party. However, your backstory or the nature of the game may prevent that; talk to your DM about it.