... I have been using the Vivitar Auto Thyristor 2800-D on camera, but so far the photographs look good but not great. They look flat.
When you shoot direct on-camera flash (i.e., point the head of the flash straight at your subject), then all the light is coming from the same direction as the camera lens. You're unlikely to have the shadows that give depth cues to the eyes, and the light is hard/harsh, so you do get that deer-in-the-headlights look so well know to P&S flash users.
If you bounce your on-camera flash (i.e., point the head of the flash at a reflective surface to diffuse the light), if you don't flag off the head of the flash with something like the Black Foamie Thing, while the majority of your light was bounced, there is still some light coming directly from the flash head towards your subject. And given the restriction your flash has of a non-swiveling head (I.e., it can only tilt up and down, it can't rotate side-to-side like most speedlights), all your light is still on-axis.
Do I need to get the flash off the camera and master zone focusing?
Zone focusing has nothing to do with changing how the direction or quality of the light from your flash.
But, given that the speedlight you chose doesn't have a head that swivels, you must get it off-camera to get the light off-axis. And getting it off camera makes it far easier to control the quality and direction of the light.
For settings, I have been using the back of the flash for suggestions. For example for ISO 400, A1 gives you F4 6-40 feet A2 F8 3-20 feet and then it also has M giving you feet and F numbers.
This is for direct on-camera flash. Back in film days, you were typically more worried about getting a decent exposure than getting good lighting, because you couldn't see what you were doing and every exposure cost you money. With digital, it's a different proposition.
This is just my opinion, but when you went for a cheapie all-manual $60 flash that's basically designed in the film era (that doesn't swivel and is probably missing 1/8 power setting), you made life a lot harder for yourself than getting a cheap Chinese TTL/HSS-capable flash with a swivel head, or saving $200 and getting a used OEM flash.
Flash can be as transformative to your photography as a lens. Budgeting accordingly might not be foolish.