The clicklists at the union room are a good place to get
some interactions done. Generally, interacting a lot yourself is
the best way to get interactions back and thus hatch and level your
pokemon faster. Chances are you won't get all your interactions
returned but plenty of us do click through our unreturned
favors.
You might want to give the pokemon on the index page a click
every once in a while as well. If you interact with them, the
eggs/pokemon in your party have a chance to show up there next.
Which will get them plenty of clicks.
Additionally, if you haven't bought the flame orb from the item shop
yet, you might wanna do that as soon as you can afford it. It'll
boost the amount of EHP your eggs gain from interactions. (The
amulet coin is also a great investment to get early on.)
Basically, what it does is, you interact with an egg, and that
gives the Egg a number of Egg Hatch Points (EHP). When the
Egg reaches full EHP (for example 5,355 / 5,355), the Pokemon
hatches. Same goes for Pokemon except that the Pokemon gains
Experience and levels up.
And how you do it is that you interact with other peoples Pokemon
and eggs (especially the ones in their party) and they will most
likely return the favor and interact with your Pokemon and eggs.