You should take a look at Windows Azure Web Sites. Web Sites offers a very straightforward way to deploy your apps to Azure without worrying about any virtual machine setup. It also has a free tier, which you can then upgrade when you feel you need more capacity, want to add SSL, or add a custom domain name.
Web Sites offers ftp access, but also has full source control integration, letting you choose between TFS, github, local git, dropbox, BitBucket, and CodePlex:
Keep in mind: With Azure, you have:
- Infrastructure-as-a-Service (IaaS), where you control everything on your virtual machines, including maintenance on the OS (whether Linux or Windows)
- Platform-as-a-Service (PaaS), where you have access to the full virtual machine and can make changes at will (via startup scripts), but you're relieved of OS maintenance
- Web Sites (which I sometimes call PaaS+ ), which provides an IIS web farm, run in Azure itself, providing continuous integration (via several source control providers), autoscale, monitoring, etc.
Web Sites also lets you spin up pre-installed software by simply selecting one from the gallery. You can, of course, select just a plain Windows Server, but you can also select CMS's (WordPress, DNN), wiki's (mediawiki), bbs's (phpBB) and others. I just grabbed this screenshot, showing a few of the gallery selections:

There are, of course, times when you need to shift to PaaS or IaaS. However: Web Sites offers a significant amount of functionality, and I believe this best suits what you're asking for.
One more thing: In case you're curious about the differences between Web Sites and PaaS (which have virtual machine templates called Web Roles and Worker Roles), take a look at this answer I posted to StackOverflow comparing the two.