To begin the Mog Safe expansion quests, you have to get a bronze bed, put it in your Mog Safe, then place it somewhere in your Mog House using the layout feature.
After the next conquest tally (which happens weekly... Sunday night/Monday morning), if you talk to your moogle, he may give you the quest. Apparently only a limited number of people receive the quest each week, and it's supposed to be based on your fame in your home town. You'll have to give him a Power Bow and a Beetle Ring to expand the safe after you get the quest.
The subsequent Mog Safe expansion quests work the same way, just with different beds/items. Check allakhazam for specific details.
As for the goobiebag quest, you need to have a certain amount of Jeuno fame to receive the quests. Jeuno fame is earned by earning fame in any of the three nations (it is essentially an average, but how precisely the average is calculated, I'm not sure). Just try to build your fame in the three nations (San o'Dria is easiest, Bastok is next easiest, Windurst is hardest, especially since Bastok has been dogging it lately in the conquest race--if Bastok isn't ranked 3rd in the conquest race, you can buy cornettes there to turn in for Windurst fame).
If you get a decent level of fame in Bastok and San o'Dria, you should have enough Jeuno fame to do the goobiebag quest. Exactly how much fame is needed, I don't know.