They do have the pinout right on the page you link, which is good.
For a bare IC, the universal rule is that you need to connect all the power and ground pins, however typically on a module they are connected by the PCB so this is less necessary.
One thing you don't know is which UART the console would be on - hopefully UART2, but could be a different one. Hopefully there is already a working U-Boot and Linux image on there.
If not, fortunately they brought out the SPI_CS0 which means you can hold the MT7688 in reset and directly write the flash chip with an external tool. Exposing SPI_CS0 is a very good decision I wish the other module vendors would make - it makes it far easier to load custom images into a product designed around the module It also looks like you have enough pins for JTAG, but that is painful.
Something to be aware of is that many MT7688 modules will not boot if you have a USB serial adapter putting voltages on the UART pins before the MT7688 gets power. This is surprisingly hard to avoid - even a large series resistor may not be enough without some clever clamping circuitry, but try starting with only the module's output (try all three UART TX's) connected to your USB serial receive - don't connect the transmit.
You can also try tripping the reset, though systems with the SPI 3 vs 4 byte issue may fail to boot in this case.
If you have a scope or logic analyzer, look for activity on the SPI pins after you release reset. With a simple USB logic analyzer you can watch it load the bootstrap part of U-Boot, and readily deduce things like the 3 vs 4 byte address issue.