There's no clear upgrade path here, you're making a technology jump of 10 years. You're effectively going to make the new system do what the old system did, test that works, then replace old with new.

If the existing system just supplies services from regular Linux packages like webserver, FTP server, etc then that's relatively simple, there will be equivalents (possibly even the same package name) in RHEL 7.

If you have third-party applications then find out if those run on RHEL 7 and any migration path the vendors offer to move the app data between the older version you're probably running now and their latest version which you'll probably need to run on the later OS.

Buy a system which has a hardware certification to run RHEL and meets your processing/memory/storage needs. Install RHEL 7, get it working, configure the same functionality as the old system, copy any data across, test functionality, schedule a cutover with a rollback plan.

Easier said than done but that's a basic workflow.