No, you may not have multiple Ambitions. The closest you can get is to get to the end of one Ambition, use the Lethean Tea Leaves (which will cost you a bit over $10 USD in Nex), and start another one. Alternatively, you could run multiple characters, each with a different Ambition. (Note that the various Ambitions are not complete, per se: each one simply stops at some point, with promises that the writers of Fallen London will create more content in the future.)
Ambitions are about investment: they're a long-running storylines that give your character a purpose. That being said, they're good fun to follow. Just pick one that appeals to you and see where it takes you.
There is now one Ambition that you may pursue in parallel with your main Ambition. It is Ambition: Enigma, and to date, it consists of solving puzzles. (Since the self-annihiliating Seeking Mr. Eaten's Name content has been disabled indefinitely, it is a sort-of replacement for it - without nearly as much pain, agony, tears, and destruction of one's property, sanity, and sense of self.) In fact, its entry-point is a puzzle; the basic clues of which have been collected. Ambition: Enigma involves references to TS Eliot, best known for the Waste Land, JL Borges, whose most famous work is The Aleph, Stamford's The Aleph in the Mirror, and Calvino's Invisible Cities and If On A Winter's Night a Traveller.