STM32 print via UART without Nucleo/Discovery
There is no problem with it. You have few options. 
The first way is to configure your UART (the process may be very straightforward if you use CubeMX) to send text and then hook-up RX and GND pins of your USB-RS232 to TX and GND pins of your board respectively. Then you can transmit your logs for example with function HAL_Uart_Transmit(). More advanced option is to redirect stdout to that UART. But it will take a lot of effort to configure and run this.

There is much better and easier way to achieve your goal and it is shown in the video. Original ST-Link V2 (and even chinese clones except the cheapest dongles) have SWO pin. You can redirect stdout of your application through this interface by means of ITM Trace technology which is a part of CoreSight debug core of your MCU. First of all you should enable SWO pin in CubeMX utility. Then you should redirect your stdout to ITM macrocells, for that purpose you can use code from this article. It will allow to do a printf-style logging. To see those logs you can use SWO viewer from st-link utility ("Printf via SWD" button). There may be a problem if you use SWO viewer from ST-link utility and debugger from your IDE simultaneously: ST-link driver won't work with two apps simultaneously. That is why I recommend to use more advanced IDE's. You can take a look at Keil uVision: it allows to use ITM and debugger capabilities  simultaneously from uVision debugger. They have evaluation licenses, so you can give it a try.

There is a third way to solve your problem and it is called semihosting. I have a very tiny experience with it and I cant recommend using it. A semihosted event halts the MCU and needs support from the debug tools to handle the semihosted operation and without debug tools attached, a semihosted event will permanently halt the MCU.