Hi Michael,
A simple approach to do away with Applets would be to use irpt/js pages under Web projects in WB. You can use SAP UI5 libraries to develop UI and jQuery to call illuminator/runner services to invoke queries/transactions.
The way you use SAP UI5 depends on what version of MII you are using, for 14.0 onwards, it comes bundled in the package.
You can use different architectures/standards to build you project, one of those suggested by Sam i.e. SAP Fiori guidelines.
Hope this helps.
Thanks and Regards,
Swaroop