How to migrate from PGSQL 9.3 to 9.6

I've never worked with Postgres, but I've been tasked with upgrading our product from using 9.3 to 9.6. Please feel free to direct me to appropriate newbie docs.

I've found the RPMS I need, but I've come across a setup file that I think I need to get a correct version of. I found references to postgresql-setup, but I have a feeling that I may need a version specific copy of that script. If I get the latest version for 9.6 will it recognize my 9.3 database and upgrade it correctly? Or do I need to edit the content to indicate that the old version is 9.3. The PGPREVMAJORVERSION reference below is what makes me wonder.

In my current db code I found a script called db_service.sh that includes these lines:
# Version 9.3.2 Devrim Gunduz <devrim(at)gunduz(dot)org>
# Add process name to the status() call. Patch from Darrin Smart

# PGVERSION is the full package version, e.g., 9.3.0
# Note: the specfile inserts the correct value during package build
PGVERSION=9.3.5
# PGMAJORVERSION is major version, e.g., 9.3 (this should match PG_VERSION)
PGMAJORVERSION=`echo "$PGVERSION" | sed 's/^\([0-9]*\.[0-9]*\).*$/\1/'`
PGPREVMAJORVERSION=9.2

# Source function library.
INITD=/etc/rc.d/init.d
. $INITD/functions
I'm not sure that this is a standard PG script or a conglomeration of other things. Is this based on the setup script or do I need new versions of both?

The current database is running on RHEL 6 and the new version will be running on RHEL 7.2. Does that affect what I need? I found an exchange in the archives about someone upgrading 9.4 to 9.5 and having problems getting the correct upgrade syntax. If you can provide that it would be greatly appreciated.

I believe I also need to upgrade our JDBC version. What is the minimum version that is compatible with PGSQL 9.6?

If these are too dumb for the general list feel free to redirect me to the novices list.

Thanks,
Ian