A suitable email client for a growing company.

I am currently involved in a company that is steadily progressing in it's field, and is expanding. So far, as it quite small, the majority of employees have been using Microsoft Outlook as their email client. Whilst many think the interface of Outlook is fine, we are now looking for an email client to use that stores emails on a cloud which can be retrieved from anywhere in the world, on any computer.
We would also like to be able to maintain our email addresses and not have to use one with the client's name in it.
A professional system and interface would be ideal.
We have a significant amount of email traffic, often with attached images.
With more users becoming involved, we are in search of a client that can fulfil our needs, and that will enable us to add and change as our company does.
Any suggestions or comments?
I am not really sure which information I should give to help, so feel free to ask away.
Thanks a lot.