How large should I make root, home, usr, var, and tmp partitions?

I wouldn't separate those partitions. That was good in the old days when you had very small disks. Sometimes though it can make sense to use separate disks for the home partition, as that is where users profiles & their data is stored, If you need to reinstall the OS their data is still there.

It also depends a bit what you are using the server for. If you are going to use it to host VM's KVM, you should assign a lot of space to those VM's. That should be on different disks than the OS. Again it is mainly because you don't want to loose the VM's when reinstalling the OS.

If you're using it for a desktop system, you can just use a single partition, but that's not ideal, unless your disk is large enough that it won't likely ever fill up.  Never let your Linux OS system partition fill up.  You will lose data on whichever partition fills up.  It's still best to partition them somewhat because of how Linux file systems work.

If you're using it for a server that has many people logging in, you really must create separate partitions.  Users can accidentally fill the disk and clobber the OS partition if it's a single partition.  Log & Temp files in Var can also do that.  It's still better to separate them.  If you don't know what to set, you can let the installer set it for you.  It should be one of the options.  https://access.redhat.com/documentation/en-us/red_hat_enterprise_linux/6/html/installation_guide/s2-diskpartrecommend-x86

I generally like to setup a swap partition so I'll create 2. 1 for the swap and 1 for the installation on a 500G drive
When I know I'll be eating space (VM's etc) I'll install an additional dedicated 3 or 4TB drive