Hello Jaydeep,
Currently, you have two choices regarding the UI:
Silverlight UI:
For now, Firefox is OK to use. I am personaly using it for Business ByDesign Silverlight version.
Only issue is with the Business Analytics work center where I need to use IE (Silverlight just crashes in Firefox in this WoC)
Chrome has some glitches with Silverlight but you can still use it by installing a plug-in.
And, yes IE is the way to go for now.
HTML5 UI:
Nevertheless, you can also log in to your tenant of ByDesign and Cloud for Customer from the HTML5 version.
Just type this address in your web browser:
https://myXXXXXX.sapbydesign.com/sap/public/ap/ui/repository/SAP_BYD_UI/HTML5/client.html
To be noticed: there is no Applicaiton and User management in the HTML5 version .... for now 
And just for your info, there will also be a Fiori client in the future. Which I am personnaly starving to see completed.
Hope this helps.
Best regards.
Jacques-Antoine Ollier