You don't need to change the location of your company. From your question it sounds like you want to keep your LLC active in Michigan, as well as in Indiana. You can keep your company in Michigan as is, and then file as a foreign entity in Indiana. Doing this will allow you to do business in both Michigan and Indiana. This should be a very simple and straighforward process.
Alternatively, you can close your business in Michigan and register in Indiana as a new LLC. But you won't legally be able to do business in Michigan anymore.
Keep in mind that you're going to have to make sure your LLC name isn't already taken in Indiana.
How you file your taxes depends on the tax structure of your LLC. Is the LLC taxed like a partnership or a corporation? If the LLC is taxed as a partnership, the taxes pass through to you as the owner. So if you are a resident of Indiana, you will pay state income taxes to Indiana. If the LLC is taxed like a corporation, the LLC pays income taxes on the profit, and you as the owner only pay income taxes on the amount distributed to you. So you may owe income taxes to both Michigan and Indiana. You should seek advice from a tax attorney or accountant.
Hope this helps!
Edit, in response to your comment:
So if I'm a website, I can only do business (i.e. complete revenue transactions or take on sales) in the state in which the LLC is registered?
Each state defines "doing business" a little differently, but in most cases, you need to have a physical presence in the state to be considered as doing business in that state. And again, the definition of "physical presence" varies from state to state. But to give you a basic example, lets say you run your business entirely from home, and sell software online to people that may live outside of your home state. Odds are you are not "doing business" in other states and should be okay. But lets say that instead of selling software you sell a tangible object and as a result you have inventory that you store outside of your home. Lets say that you store some of your inventory in Indiana and store the rest in California, possibly for quicker shipping. Now you have a physical presence in both Indiana and California, so you will have to register your company as a foreign entity in California.
If your situation is like the first example I gave, you will have to register your business in Indiana, but you won't have to register in every state. As for your business status in Michigan, you can choose to keep it in good standing if you'd like, or you can end it. Either way is fine.
Each situation is a little different and each state has their own laws, so you should speak to a lawyer about your unique situation. Hope that clarifies things for you!