I don't think my SD-Card image would work on the older PI Zero without WIFI... I thought it had a substantially different system-on-a-chip... so stay away from that.
PI Zero W,
PI 3 B,
PI 3 B Plus
All versions of the Pi Zero and Pi 1 (A, B, A+, and B+) use a BCM2835. Depending on which pins tipi uses, the cable you are using to connect to the pi may not be fully compatible with the model A and B.
The Pi 2 B used the BCM2836 until October 2016. (ARMv7) If you buy a Pi 2 B after that, you're likely to get a v1.2 board which uses a BCM2837. Broadcom stopped making the 2836, so the new v1.2 Pi 2 model B was basically a Pi 3 model B without wifi/bluetooth. Didn't cost any less, but there were some users who needed the version without radios, so RPF kept it around.
The Pi 3 B, as mentioned, uses the BCM2837. The BCM2837B0 is a slight mod to the 2837 with the heat spreader allowing it to clock a little higher without quite so much heat generation. That combined with the new support chips on the Pi 3 B+ make it clock faster and have better WiFi and Ethernet. Technically the BCM2837 is a 64-bit ARMv8 device, but … we use them on the Pi as 32-bit ARMv7(s) devices because 1GB RAM sucks for 64 bit twice as much as it does for 32.
I wouldn't recommend a Pi 1 A/B, but the A+/B+ should just work for you. And if they work for you, probably the older 2 B+ will work without much modification aside from verifying what speed it wants to operate at.
Clear as mud, eh? If you have a Pi Zero without WiFi, you'd probably want either an Ethernet dongle supported by Linux (no, Android support doesn't mean it will work for those USB-OTG models) or a USB WiFi adapter (again, must be Linux supported). But by the time you're doing all of that, really, why don't you just get yourself a Pi Zero W from Greg with the convenient header already installed, or pick up a Pi with the networking option you want built right in? The Pi is the cheapest part of getting TiPi set up and running.