First it will be uploading data to the cloud so that people can use the Access front-end with an SQL Server Online backend. That's relatively easy to do. After that, I'll be building a web-based portal for the database so people can use their web browsers to view/edit data.

Is an SQL Server online backend faster than a sharepoint tables backend, which I find very slow...?

Generally, yes, but it depends on a lot of factors: the location of the server, quality of the network traffic between the two of you, etc.