Azure is a great place to have your app, but there are some considerations you need to know before start migrating it.
Azure Websites and Hosted Services are really trivial to deploy. With
Visual studio you generate the package and simply upload it. Then you
have a Development environment to check it. If it's ok for you, swap
ips. If it's not ok for you, upgrade again.
Your instances have some properties that could be annoying. For
example, you cannot be sure about your IP. Then if your app works
with some provider using IP restriction, you will need to figure out
how to proceed.
More considerations. Your "server" could be reimaged at any moment.
If you store something on the local disc, that file could go away at any moment.
Azure works very nice if you have at least 2 instances or more for
each website. Maybe your app is not prepared for that. The first step
will be managing the sessions with the appFabric. Is really
easy, just a change on your web config. Be careful because this
session state doesn't work exactly as the "old one". You cannot store
non-serializable objects (should be easy to adapt) or a very large objects (more than 8MB).
If you are going to develop something from zero, I suggest you to start into azure from the beginning. The reason is simple: it's really cheap to start and you will not pay serious money until the app have lot's of visits. It's also very cheap to setup a SQLAzure and a storage account. One you have all in place, it's easy to add more instances or scale up.
Example:
Imagine you have an idea and you wish to show up to some possible investors.
You start setting up a little SQLAzure database (1GB ), $9,99 monthly.
Then you build a site and you put 2 extra small instances, $18,72 monthly.
Let's say you need 100 GB of space (images, backups, ...), $12,50 monthly.
At his point, you have all in place to start your business paying less than $50 monthly.
If you site have exit and the visits starts to come, you change your instances for small instances (it's really dangerous to have production environment with extra small instances, because do not have cpu reservation). Then you change the extra small cost ($18,71) up to $57,60. Maybe you need more space to that SQL Azure? etc...
prices calculated from here: http://www.windowsazure.com/en-us/pricing/calculator/?scenario=web .
Those are few tips, there is a lot more. My advice is to start a trial account and play with it.
Final advice: Its very easy to solve everything just purchasing more resources. Sometimes you need to refactor and optimize your code. If you simply add more resources each time you have a problem, you could end with a huge bill and a very messy code.
Hope it helps!