I have one of those displays, also purchased from a discount seller on ebay. It's a clone of a clone of Adafruit's display, but it works identically. You can find setup instructions that work here:
You just need to get the waveshare35a dtoverlay file and reference it in /boot/config.txt.
The display will work on the RPi Zero, B+, 3B (I have those 3 types of boards and tested it on all of them). This particular display may disappoint you if you're going to use it for high framerate activities like video playback or gaming. The SPI interface on the board talks to relatively low speed shift registers and maxes out at 12-16Mhz. For a 320x480x16-bit display, this means frame rates of 10fps or less. If you don't mind this limitation, then it's a decent little display. When plugged in and no software to talk to it, it will just show a blank white screen. The backlight LED is just connected to a logic level and cannot be dimmed.
Once you have it set up with the dtoverlay, it still won't display anything because the dtoverlay creates a virtual frame buffer at /dev/fb1. If your software doesn't know to send it's output there, you can configure Linux to use it as the main display. You can also just talk directly to the display and bypass the normal 'Linux' way of dealing with framebuffers. I've published C code which talks directly to this type of display: