In terms of the origin world, it's not really that difficult to get through all of the quests on the planet before you finally are able to leave. I actually do recommend that people stack as many quests onto their quest log since it will save a lot of time running back and forth and you'll end up finishing multiple quests at once.
In no time you'll find that you've completed all of the quests on the planet, have reached lv10-12, are ready to get your advanced class on the fleet station along with your crew skills, and are prepared to take on the first flashpoint with friends and continue your adventure without having to go back to your origin world.
Stacking a tonne of quests shouldn't be an issue and it's probably for the best if you do it early, even if you're new to the concept like I was not long ago, since it will help you learn how to manage it more quickly. Better to learn earlier in the game than later where things get even more complicated and you're trying to learn about other aspects.
As for companions, just note for everyone that each class gets their companion at different times, for example the Trooper gets his/her companion by the time they're ready to leave their origin world while the Sith Warrior gets their companion in the middle of doing their story quests on the origin world.
So yeah, my advice for the origin world would be to do everything that's available there and get the most out of it as possible. Don't skip anything cause BioWare has done a lot of fine tuning on those worlds to ensure that when you leave you're prepared fully for everything that is to come.