When I'm flying VOR-VOR I pay attention to the flight plan and the headings for each VOR. As I get within a couple miles of a VOR point I'll set my heading bug to the current heading, change from NAV to HDG on the auto pilot, switch NAV1 to the new frequency and change the NAV heading indicator to the appropriate heading in my plan. As I cross the VOR point the NAV1 indicator will move till I'm on target and as I get real close I then switch autopilot from HDG to NAV and head off to the new VOR.
This does not work with the way points that lay between VOR points. Sometimes when setting up a flight plan on airways it will make turns that are not on VOR points but on other way points. This really only works with the GPS, BUT if you know the headings and the distance from or to the VOR points you can use the HDG setting and the DME to fly those legs.
Old school nav was fly a heading from a certain point at a certain altitude at a certain speed for a certain period of time. Then you had to figure in wind speed and direction.
I remember my flight instructor/family friend always had a slide ruler and a series of compasses and measuring tools in his flight bag along with all his sectionals. He was a math professor and loved doing things the old way when he wasn't flying by view.

I don't recall him ever using auto pilot or ever setting his NAV radios. He would set COM1 to the frequency of where we were flying from and COM2 to the destination and almost never took it above ~8K AGL. Heck, usually we never broke 5K AGL.