What is the reason for not wanting to register in your home state? Is it to save on the CA state fees? If so, your not going to like my answer.
Since you will be operating your business from your home that means your home becomes your place of business, which means you will be "doing business" in CA. Each state has their own definition of what "doing business" in that state means, but most states consider working out of your home as doing business in that state. This means that even if you register your business in another state you will still have to register in CA, and pay CA fees. The best way around this is to become a resident of another state.
Another thing to keep in mind is your company's tax status. You indicate that you want to form an LLC, but you didn't tell us if the company will be a pass-through entity or if it will be taxed like a corporation. By default LLCs are treated as pass-through entities. A pass-through entity is one that does not pay taxes itself, but instead its profits are passed through to its owners, and the owners pay the taxes on their individual tax returns. If your LLC will be a pass-through LLC, then you will also have to pay CA taxes on your profits. See Simple Overview of LLC Pass Through Taxation for more.
For these reasons, I usually recommend startups start by registering in their home state. Once you start making so much money that taxes start becoming an issue (and you can afford a good tax attorney), then revisit this idea.
To answer your other questions, as an online business you won't have to register in every state as long as you don't have a physical presence in those states. See Forming single member LLC for internet website and Which state should I choose for incorporation for a new online sales business for more on this topic.
Moving to NY in the future doesn't really change much right now. You can register in CA now, and then when you move to NY register your business there. See Startup Tax Question - I moved, company didn't for more on this.